COMPLEXITY, COMPETITION AND GROWTH:
KEY IDEAS FROM ADAM SMITH, MODELED USING AGENT-BASED
SIMULATION
by
Nathanael Smith
A Dissertation
Submitted to the
Graduate Faculty
of
George Mason University
in Partial Fulfillment of
The Requirements for the Degree
of
Doctor of Philosophy
Economics
Director
Department Chairperson
Program Director
Dean, College of Humanities
and Social Sciences
Date:
Summer Semester 2011
George Mason University
Fairfax, VA
Complexity, Competition and Growth:
Key Ideas from Adam Smith, Modeled Using Agent-Based Simulation
A dissertation submitted in fulfillment of the requirements for the degree of Doctor of
Philosophy at George Mason University
By
Nathanael Smith
Master of Public Administration / International Development
Harvard University, 2003
Dissertation Director: Charles K. Rowley, Professor
Department of Economics
Summer Semester 2011
George Mason University
Fairfax, VA
Copyright 2011 Nathanael Smith
All Rights Reserved
ii
ACKNOWLEDGEMENTS
I wish to acknowledge the assistance of my dissertation committee and especially the
extraordinary devotion of Charles Rowley in helping me finish this dissertation. His
attentive support was indispensable to the realization of this unique and ambitious
scholarly project.
iii
TABLE OF CONTENTS
List of Figures
Abstract
Introduction
Chapter 1 The Invisible Hand Re-Loaded: New Microfoundations for Price Theory
Appendices
Bibliography
Chapter 2 The Division of Labor is Limited by the Extent of the Market
Appendices
Bibliography
Chapter 3 Bayesian Skill Reputation Systems: A Contribution to ‘Epistonomics’
Bibliogaphy
iv
Page
v
viii
1
14
91
103
106
209
252
255
330
v
LIST OF TABLES
Table 1: There is no Walrasian equilibrium in the Triangle Economy.................................... 141
vi
LIST OF FIGURES
Figure
Page
Figure 1: The "core" in an Edgeworth box model .................................................................................. 21
Figure 2: Structure of the simulation (simplified UML)......................................................................... 30
Figure 3: How the empiricist algorithm works with constant AC .......................................................... 34
Figure 4: How the empiricist algorithm works with U-shaped AC ........................................................ 35
Figure 5: How the empiricist algorithm works with falling AC ............................................................. 36
Figure 6: Duopolistic competition with constant AC has a cyclical character ....................................... 38
Figure 7: Cyclical peaks vary under duopolistic competition................................................................. 40
Figure 8: Price converges to costly swiftly in empiricist competition, as n increases ............................ 42
Figure 9: With U-shaped AC, competitive cycles in duopoly are more irregular .................................. 45
Figure 10: Price fluctuates within a narrow band above cost in duopoly with U-shaped AC ................ 46
Figure 11: With U-shaped AC, more firms than n=2 do not lower the price much................................ 47
Figure 12: With zero transactions costs, empiricist competition cannot discipline prices...................... 49
Figure 13: Price falls with transanctions costs until close to zero .......................................................... 50
Figure 14: Empiricist competition does not make firms produce at efficient scale ................................ 57
Figure 15: Profits vary within a wide band around the threshold profit rate .......................................... 59
Figure 16: Firms’ prices track each other closely ................................................................................... 60
Figure 17: Sales for individual firms are quite volatile, and generally above efficient scale ................. 61
Figure 18: Free entry with profit threshold of 5% leads to prices about 50% above minimum AC ....... 62
Figure 19: Profits are volatile, generally below the entry threshold ....................................................... 63
Figure 20: The simulation with free entry generates fewer than optimal firms ...................................... 65
Figure 21: Under falling AC, firm production levels exhibit moderate volatility .................................. 68
Figure 22: Profits stay a bit under the threshold, with occasional large deviations ................................ 69
Figure 23: Price follows a cyclical pattern, with alternating price-cutting and shakeout phases............ 70
Figure 24: Sales of individual firms are highly volatile.......................................................................... 71
Figure 25: Free entry under falling AC leads to fairly stable competition and price discipline ............. 74
Figure 26: What does a downward-sloping supply curve mean? ........................................................... 75
Figure 27: With falling AC and free entry, a downward-sloping supply curve is possible .................... 78
Figure : Falling AC does not necessarily lead to monopoly ............................................................... 80
Figure 29: Empiricist firms make more profits than Randomizer firms ................................................. 84
3T
3T
3T
3T
vii
Figure 30: Empiricist firms generally beat Exploratory firms, too ......................................................... 86
Figure 31: Between Empiricist firms and Price Taker firms, the contest is close .................................. 87
Figure 32: Empiricist vs. the Genetic Algorithm is another close contest .............................................. 88
Figure 33: With a GA algorithm, price also converges to cost fairly quickly ........................................ 89
Figure 34: Without a zero bound, inventories may turn negative while a retailer finds its way to
equilibrium ............................................................................................................................................ 146
Figure 35: The stockout pricing policy keeps inventories above zero and eventually equilibrates ...... 148
Figure 36: Barter versus money ............................................................................................................ 154
Figure 37: Equilibrium size of the retail sector..................................................................................... 156
Figure 38: The near efficiency of a Howitt-Clower economy .............................................................. 157
Figure 39: The persistent barter character of a Howitt-Clower economy ............................................. 159
Figure 40: Barter vs. money.................................................................................................................. 160
Figure 41: How the Howitt-Clower model (with money) converges to Walrasian prices.................... 163
Figure 42: Stockout constraints in consumption-vector space.............................................................. 168
Figure 43: A feasible optimal consumption vector under stockout constraints .................................... 169
Figure 44: With jobout constraints, the wage becomes a function of hours worked ............................ 171
Figure 45: Technology .......................................................................................................................... 174
Figure 46: The emergence of market-oriented industries ..................................................................... 175
Figure 47: Industry sales are quite volatile ........................................................................................... 177
Figure 48: The home production option makes prices in Industry 5 less volatile, sales more so ......... 178
Figure 49: Demand for Good 1 is less elastic, making price more volatile, sales a bit less so ............. 179
Figure 50: “From subsistence to exchange,” or, how market labor displaces home production .......... 180
Figure 51: How the shift to market employment raises utility.............................................................. 181
Figure 52: Utility tracks market employment closely ........................................................................... 183
Figure 53: Flexibility of specialization ................................................................................................. 184
Figure 54: A larger market leads to higher utility................................................................................. 187
Figure 55: Labor diversifies as the population grows ........................................................................... 189
Figure 56: Consumption is diversified but falls by less than labor because fixed costs are reduced.... 190
Figure 57: Small economy goes for "low-hanging fruit," larger economies explore more of the
technology space ................................................................................................................................... 192
Figure 58: A technology for which some goods require capital to produce ......................................... 199
Figure 59: Smithian growth without capital ......................................................................................... 200
Figure 60: Industrial diversification...................................................................................................... 201
Figure 61: What the economy without capital produces....................................................................... 202
Figure 62: A rise in the savings rate leads (eventually) to steady growth ............................................ 203
Figure 63: The savings rate and the growth rate are positively correlated ........................................... 204
Figure 64: Capital accumulation leads to rising quality of goods......................................................... 205
Figure 65: High savings leads to a growth take-off .............................................................................. 206
Figure 66: Savings rates boost growth .................................................................................................. 207
Figure 67: Nowhere else to go on the quality ladder ............................................................................ 208
Figure 68: The agent's problem............................................................................................................. 218
Figure 69: Cycles .................................................................................................................................. 220
viii
Figure 70: Acquiring the target good .................................................................................................... 222
Figure 71: A moderate pessimistic bias allows investors to make the most money ............................. 285
Figure 72: The economy converges to a steady state in which GDP per captia is nearly constant
over time ............................................................................................................................................... 292
Figure 73: The distribution of wages among those working in a profession is highly skewed ............ 294
Figure 74: Highly paid specialists all enjoy far-reaching reputations................................................... 296
Figure 75: Only skills that are truly of good quality acquire good reputations .................................... 297
Figure 76: There is persistent wage dispersion, which appears to be largely unrelated to age............. 299
Figure 77: Reputation reach is associated with higher labor income, but not very strongly ................ 301
Figure 78: GDP per capita eventually falls with higher population, but it initially rises...................... 303
Figure 79: A larger population raises productivity through a finer division of labor ........................... 304
Figure 80: The raw data on labor and capital income ........................................................................... 307
Figure 81: Capital and labor income tend to rise and fall together ....................................................... 308
Figure 82: Capital income is more volatile in the very short run than labor income ............................ 309
Figure 83: The more opportunities there are in the economy, the larger is capital's share of income .. 311
Figure 84: When agents consume more of their wealth each turn, GDP is lower ................................ 313
Figure 85: Pessimism on the part of capitalists reduces GDP per capita .............................................. 315
Figure 86: A climate of pessimistic expectations acts as a form of collusion, raising capital's share
of income .............................................................................................................................................. 316
Figure 87: A rise in pessimism (a decline in "animal spirits") leads to a drop in GDP per capita ....... 318
Figure 88: The rise in pessimism raises the capital share of income by about 10% ............................. 319
Figure 89: Capital income falls a bit when pessimism rises ................................................................. 320
Figure 90: Pessimism leads to a rise in wage inequality....................................................................... 321
Figure 91: Pessimism, arising endogenously, has a clear central tendency .......................................... 324
Figure 92: When all skill information is suddenly lost, GDP per capita plunges, but recovery is
swift ...................................................................................................................................................... 325
Figure 93: GDP per capita falls by about 20% when agents learn to conceal their failures with
50% probability..................................................................................................................................... 328
Figure 94: Labor income stays at exactly the same level before and after the opportunity for
failure concealment appears.................................................................................................................. 329
Figure 95: Capital income drops sharply when agents learn to conceal failure.................................... 330
Figure 96: When agents are able to conceal their failures, employers become slightly more
pessimistic ............................................................................................................................................. 331
Figure 97: GDP per capita is an increasing function of the information available to agents................ 332
Figure 98: Labor income is constant for visibility of failure over 30-40%, but falls when failure
concealment depletes capital................................................................................................................. 333
Figure 99: Capital income falls off as failures become less visible ...................................................... 334
Figure 100: Employer pessimism responds weakly to declining visibility of failure, and then
becomes confused ................................................................................................................................. 335
ix
ABSTRACT
COMPLEXITY, COMPETITION, AND GROWTH: KEY IDEAS FROM ADAM
SMITH, MODELLED USING AGENT-BASED SIMULATION
Nathanael Smith, Ph.D.
George Mason University, 2011
Dissertation Director: Professor Charles K. Rowley
This dissertation uses agent-based simulation to study markets in ways that depart from
the Walrasian tradition, and to vindicate Adam Smith’s beliefs about the power of the
division of labor to enhance productivity, which mainstream economics has neglected
because Walrasian equilibrium is incompatible with nonconvexities such as fixed costs.
The first article, “The Invisible Hand, Reloaded,” studies a market in which firms in a
commodity market try to maximize profits by estimating demand via an OLS regression,
while customers choose the lowest-price firm, but face random firm-specific transactions
costs. I call this “empiricist competition.” Near perfect competition emerges with very
few firms (e.g., n=3), and the result cross-applies to the free entry case, to U-shaped
average costs, and even to (gently) falling average costs, in which case empiricist
competition gives rise to a downward-sloping supply curve. The second article, “The
Division of Labor is Limited by the Extent of the Market,” vindicates the thesis of
Chapter 3 of The Wealth of Nations by building a market of decentralized retailers,
x
following Howitt and Clower (2000), and then equips agents with avoidable-cost
production functions, taste-for-variety utility functions, and techniques to sift through
hundreds or thousands of corner solutions to find the optimum behavior when they face
buy-sell price spreads and stockout and “jobout” constraints on what they can buy or sell
at each price. This gives rise to endogenous specialization, which is compatible with
competition yet causes GDP per capita to rise indefinitely with population growth or with
the accumulation of “capital” (foregone consumption, subject to diminishing returns and
depreciation, which augments labor). What emerges is an interpretation of technological
change, not as new discoveries a la Romer (1990), but as the exploration of an alreadyknown technology space which requires sufficient labor-cum-capital to explore. Unlike
Romer (1990), this interpretation of technology is a candidate to explain the wealth and
poverty of nations. The third article, “Bayesian Skill Reputation Systems, presents a
model where agents are endowed with skills whose quality is unobservable and
opportunities arrive each turn which can only be exploited by certain skills, I show how
agents can use Bayesian updating to derive pretty accurate knowledge about the quality
of skills. However, the Bayesian skill reputation system quickly fails when agents can
conceal past failures, which suggests a reason why, as Granovetter (1983) showed,
networks of “weak ties” are so important for finding jobs, and why employers are
suspicious of gaps in resumes.
xi
INTRODUCTION
Adam Smith’s Wealth of Nations is the founding text of economics. Smith had
forerunners of course, but they are rarely revisited except for historical curiosity’s sake,
while in the Wealth of Nations, the worldview of economics was born fully armed and
grown, like Athena from the head of Zeus. The modern economist will draw graphs of
which Smith knew nothing, and which distill into a few lines abstract concepts that Smith
struggled to convey through hundreds of pages of prose. But what the graphs teach,
Smith understood.
Adam Smith had two big ideas. The first, which is developed in Chapters 1 to 3, was the
division of labor, which, according to Smith, is the main reason that labor in advanced
economies like 18th-century Britain (or, a fortiori, the contemporary United States) is
more productive than labor in less advanced economies. Smith cites several reasons for
this including (a) the greater skill that specialists attain through constant occupation with
1
the same narrow task, (b) avoiding the loss of time and concentration that occurs when an
individual has to keep switching between tasks, and (c) the tendency for specialization to
spur the invention of labor-saving devices. The division of labor is made possible by
humans’ propensity to exchange, and is limited by the extent of the market.
Smith’s second big idea, which is developed throughout the book, is that the “invisible
hand” of competition coordinates supply and demand through the price mechanism.
Without central planning and coercion, even without the need for virtuous regard for the
interests of others, the efforts of individuals in a complex economy are directed to the
satisfaction of one another’s needs and wants. When a good is too scarce, its price rises,
causing buyers to economize, and spurring sellers to increase their supply, until the
market clears. When a good is too abundant, the price falls to clear the market. With the
price mechanism allocating labor efficiently, “natural liberty” is compatible with
economic prosperity. Smith’s two ideas depend intimately on one another in informing
his worldview and that of economics generally. Without the invisible hand, the division
of labor argument would suggest the need for a caste system or a central planner.
Without the division of labor, to oversimplify slightly, there is nothing for the invisible
hand of the market to do, no exchange for it to mediate.
Later economists have followed Smith’s lead in devoting more of their attention to
competition, the price mechanism, and how the market coordinates supply and demand,
than to the division of labor. The reason for this is probably that the division of labor (if
2
not its degree of importance) is rather obvious, while the workings of the price
mechanism are counter-intuitive and need more explaining. Smith himself still wrote of
the “natural price” of things, even though from today’s perspective Smith’s framework
renders that notion meaningless, for it shows that prices have no natural basis but emerge
from the interaction of supply and demand. This became clear when Smith’s ideas were
further refined by economists of the “marginal revolution” like Alfred Marshall, Karl
Menger, and Leon Walras. Walras’s conceptual framework, in particular, has been the
dominant influence on how the invisible hand idea is understood today.
In the Walrasian model of markets, an “auctioneer” finds the price vector which makes
supply equal to demand for all goods, i.e., “clears the market.” Agents have utility
functions and endowments, and/or production functions, and are able, given a price
vector, to arrive at definite decisions about the optimal amount to make, buy, or sell.
Since agents’ decisions are functions of prices, individual demand and supply functions
can be derived from the agent optimization problems, and the aggregate demand and
supply functions are summations of these individual demand and supply functions. The
auctioneer finds the market-clearing price vector through a process of tatonnement or
‘groping.’ Since every Walrasian equilibrium is a Nash equilibrium – if in a population
of n agents, n-1 make exactly the offers prescribed by the Walrasian equilibrium, it is in
the interests of the nth agent to do so as well – the Walrasian auctioneer acts, so to speak,
as a matchmaker rather than a central planner, and does not need to use coercion. At the
3
same time, the Walrasian equilibrium is demonstrably Pareto-efficient. Adam Smith’s
vision, in which prosperity and natural liberty are compatible, is ratified.
The Walrasian model’s influence in economics is pervasive. The Walrasian auctioneer is
hardly ever mentioned, but when terms like “price-taking,” “competitive,” and
“equilibrium” are mentioned in the exposition of a formal model, they are a signal that at
some point the theorist, having described the utility and production functions and solved
maximization problems to derive supply and demand, will solve a system of equations to
determine the market-clearing price vector, thus playing the role of the Walrasian
auctioneer. The Walrasian model, as refined by many others culminating in Arrow and
Debreu (1954), is the implicit warrant for the mathematical formalization of economics in
recent decades. Since the Walrasian model is a formalization of the invisible hand idea,
it has tended to favor the influence of Adam Smith as against Karl Marx or even J. M.
Keynes. But it has also eclipsed Adam Smith’s other big idea, the idea of increasing
returns arising from the division of labor, through gains from specialization and trade.
The most obvious critique of the Walrasian model is that it lacks realism. Needless to
say, there is no such person as a Walrasian auctioneer. He is personification of market
processes, of “the invisible hand,” which is as much as to say that he is a placeholder for
some theory or theories that would explain why, rather than merely postulate that,
markets clear. Unfortunately, the assumption of “price taking” assumes away the very
behavior – setting prices – by which agents might find their way to (approximately) a
4
Walrasian solution through a decentralized process. But if price-setting is allowed,
agents can try to exploit market power, offering less than what would satisfy their
marginal conditions at the market price, in order to manipulate terms of trade in their own
favor. Only when the number of agents is “infinite” does price taking become strictly
rational. Also, the assumption of “price taking” is closely related to the rather
embarrassing assumption of “perfect information.” In order to solve his maximization
problem, a neoclassical consumer needs complete knowledge of what goods are available
and at what prices. In a complex economy, this is not very plausible, and it is one reason
for unrealistic predictions of the Walrasian model such as the absence of buffer stocks or
unemployment.
I would not want to overstress the Walrasian model’s lack of realism, however. It is
quixotic to hope for explanation without simplification. No doubt the Walrasian model
sometimes leads economists to be complacent, overestimating the economy’s tendency to
efficient self-regulation. But economists know that neoclassical models make idealized
assumptions, and they are often receptive, probably too much so, to arguments that begin
by showing that they are empirical deviations from rationality or competitive markets and
then diagnose “market failures.” (Just because economists sometimes fail to secondguess markets when it turns out, ex post, that they should have done so, does not mean
that they would have made these calls correctly if they had had less confidence in
Walrasian market efficiency.)
5
All three papers in this dissertation will buttress some key Walrasian assumptions. The
first paper shows that nearly perfect competition can emerge in industries with a small
number of competing firms, and that it is consistent with U-shaped average costs and
even with mildly downward-sloping costs. The second paper shows that approximately
Walrasian prices, and efficient markets, can emerge in an economy where exchange is
mediated by many decentralized traders earning margins of private profit, instead of by a
central auctioneer setting prices. The third paper shows that when the quality of skills is
not observable but performance outcomes are, the market can be fairly effective in
generating information about skill quality by a process of induction and information
pooling based on Bayesian updating.
Instead, my main complaint about the Walrasian model of markets is that it lacks
generality. The Walrasian model can be extended to any number of n 1 goods and n 2
agents, provided that n 2 >n 1 , to any initial distribution of endowments, and to a wide
variety of production and utility functions. Arrow and Debreu (1954) proved the
existence of Walrasian equilibrium in an integrated economy with production,
consumption, and a circular flow of income. But to make this work, they had to assume
that the set of “all possible input-output schedules for the production sector as a whole…
is a closed convex subset of Rl containing 0,” which “implies non-increasing returns to
scale.” Without the assumption of non-increasing returns to scale, Walrasian equilibrium
typically fails to exist.
6
Perhaps the simplest example of an economy with no Walrasian equilibrium consists of
three agents, A, B, and C; two goods, x and y; endowments x=1 and y=1for each agent;
and utility functions u=
xi 2 + yi 2 . In this economy, no price vector clears the market,
i
i.e., there is no Walrasian equilibrium. Most economies with nonconvexities in the utility
or the production functions have no Walrasian equilibrium. And the economy as Adam
Smith conceived it is full of nonconvexities, such as the fixed costs of learning a skill or
starting a task. Thus, even as the Walrasian model elucidates one of Adam Smith’s big
ideas, it has to assume away the other one. To get the invisible hand, we have to abandon
the division of labor.
The influence of the Walrasian model has led economists systematically to neglect
increasing returns and gains from the division of labor, simply because this requires
allowing nonconvexities at the micro level which are inconsistent with Walrasian
equilibrium. The outstanding example of this neglect is the Solow or neoclassical growth
model. Positioned for decades as the canonical model of economic growth, the Solow
model assumes a constant-returns-to-scale aggregate production function, thus in effect
dismissing Adam Smith’s ideas about gains from the division of labor. Solow hardly
seems to be aware that he is doing this, for he writes:
About production all we will say at the moment is that it shows constant returns to scale. Hence
the production function is homogeneous of first degree. This amounts to assuming that there is no
scarce nonaugmentable resource like land. Constant returns to scale seems the natural assumption
7
to make in a theory of growth. The scarce-land case would lead to decreasing returns to scale in
capital and labor and the model would become more Ricardian. (Solow, 1956, pp. 67)
It is striking that Solow felt the need to defend his constant-returns assumption against
possible arguments for diminishing returns, but his dismissal of the insight most
enthusiastically stressed by the founding father of economics did not even merit a
mention. Later textbook writers, however, have felt the need to point out that gains from
division of labor, at the margin, are assumed away. Thus David Romer writes:
The assumption of constant returns to scale can be thought of as a combination of two separate
assumptions. The first is that the economy is big enough that the gains from specialization have
been exhausted. (Romer, p. 10)
But it will not do to ignore the division of labor. There is too much evidence that it
exists, and matters. Henry Ford wrought a cost revolution in automobiles by placing
workers on an assembly line doing simple tasks repetitively. This is the iconic case of a
phenomenon that occurs over and over again, namely, that dramatic improvements in
efficiency can be achieved through capital investment and intensification of the division
of labor, but only if the scale of production is sufficiently large. Increasing returns is also
consistent with the stylized facts of economic development. As Sachs, Gallup and
Mellinger (1999) have shown (and as Adam Smith emphasizes in Chapter 3 of The
Wealth of Nations), access to the sea has long been an important determinant of
development, because water-carriage has long been the cheapest mode of long-distance
8
bulk cargo transport, and so serves to facilitate trade and the division of labor. And as
Fujita, Krugman, and Venables (2001) have argued, the very existence of cities becomes
hard to explain if increasing returns is assumed away. If production is equally efficient
no matter how small its scale, we might as well produce everything at home, leading to
“backyard capitalism.”
All three of the papers in this dissertation, in different ways, reconcile competition with
some sort of increasing returns or division of labor. In the first paper, I develop a model
that generates near perfect competition in the standard constant-returns case, and then
show how it can be extended, with no other changes, to an economy in which the cost
curve is downward-sloping, and give rise to a downward-sloping supply curve, a
phenomenon long regarded as impossible by economic theorists. In the second paper, I
develop an economy in which trade is mediated by a decentralized retail sector, and show
how a larger population leads agents to pursue more narrowly specialized production
strategies, and thus raise utility both by reducing fixed costs and by getting access to a
wider variety of goods. In the third paper, skills and opportunities are supplied
exogenously, but a larger population gives agents access to a wider variety of skills
which they can hire to exploit opportunities, and also facilitates the discovery of skill
quality.
Now, economic theorists have suspected at least since Stigler (1951) if not since Young
(1928) that the refinement of equilibrium theory was leading to assumptions that
9
contradicted Smith’s ideas about the division of labor and increasing returns, and that a
model reconciling Smith’s big ideas would be invaluable. Has no one done it, and if not,
how is it possible that a mere graduate student has come up with not one but three ways
of accomplishing this rare feat? The answer to the first question is, sort of. There are a
few extant models that bring in the division of labor in the context of markets that are in
some sense competitive. They rely on some very special and strange assumptions, e.g.,
that all contracts are signed at time t=0 and perfectly enforced thereafter; or that all goods
are symmetric and all markets are “monopolistically competitive.” These models are
hard to interpret or test and have not been very influential. To the second question, the
answer is that I am using a novel technique, agent-based simulation.
It is puzzling that agent-based simulation has not had more impactin economics. One of
the main desiderata of economic theory, methodological individualism, is perfectly
satisfied by agent-based simulation, which models individuals as computational objects
which contain data and possess behaviors or modes of action, and which sequentially
activates these agents, with results emerging from their behaviors and interactions.
Another of the desiderata of economic theory, equilibrium, is almost never strictly
satisfied by an agent-based model, but agent-based results often give rise to roughly
steady states or quasi-equilibria, where the major variables of the system continue to
fluctuate but exhibit approximate stability over time. Since economists know well that
“equilibrium” results are at best an approximate description of the world, it is hard to
10
believe that the difference between an equilibrium and a quasi-equilibrium is a major
barrier to acceptance of such results.
Lehtinen and Kuorikoski (2007) argue that “economists shun simulation” because “the
deductive links between the assumptions and the consequences are not transparent in
‘bottom-up’ generative microsimulations.” While this is probably the right diagnosis, the
truth is that a lot of mathematical models are surely opaque to most economists (though
they might be reluctant to admit it) while in some agent-based simulations the links
between a model’s structure and its results are easy to see intuitively. Two other reasons
may be (a) that most economists do not have the skills to make or the conceptual toolkit
to understand agent-based simulations (yet), and (b) that economists have little
motivation to appraise agent-based simulation as a methodology, because they have not
seen agent-based simulation generate impressive new insights about the economy (yet).
Two exceptions are (a) Thomas Schelling’s model of segregation (Schelling, 1971), and
(b) Robert Axelrod’s demonstration that tit-for-tat tends to be the most successful
strategy in the repeated prisoner’s dilemma game (Axelrod, 1981) against a wide variety
of other strategies, but these results, though widely accepted, have been only narrowly
applied in economics.
Another problem with agent-based simulation is that it is very hard for computational
agents to meet the standard of strict rationality that economists like to impute to the
actors in their models. Computational agents can be highly logical and they calculate
11
accurately, but it is hard to teach them to adapt to new situations through original
deductive reasoning. Now, it can be argued that real people do not meet the standard of
strict rationality, either, and that economists ought in any case to drop this assumption in
favor of assuming patterns of behavior that are empirically observed. For the most part, I
do not support this approach. I believe that we humans simply know by introspection
that we are, and/or aspire to be, in an important degree, rational, and a judicious
application of the assumption that humans are rational can yield insights with
considerable precision and generality, whereas non-rational patterns of behavior observed
in experiments tend to be vaguely described and situation-specific.
On the other hand, the assumption of “perfect information” is wildly false, and it must be
understood that there are many situations in which people just do not have the
information they need in order to make a strictly rational decision. Worse, in order to
implement a high standard of rationality, economic modelers not only need to assume
that agents have lots of information, but also need to make their model worlds quite
simple. Even agent-based simulations with a simple model structure tend to give rise to
results that are quite complex, making it impossible for agents in the simulations to
determine strictly optimal strategies.
In the three simulations of this dissertation, I have gone to considerable lengths to make
agents as rational as possible. Firms in “The Invisible Hand, Reloaded,” use OLS
regressions to estimate demand. In “A Bayesian Skill Reputation System,” agents apply
12
Bayes’ Law in order to update their beliefs in response to new evidence. Agents in “The
Division of Labor is Limited by the Extent of the Market,” which come closest to
satisfying the rationality criterion, strictly maximize an instantaneous utility function,
even though it may require a directed search through thousands of “corner solutions.” In
all three papers, however, some behaviors have to be governed by rules of thumb. Thus,
in “Division of Labor,” agents do not maximize an intertemporal utility function.
Instead, money is simply included in their instantaneous utility functions, as if they took
pleasure in holding money per se, though the interpretation of this should be that they are
adopting a rule of thumb which will benefit them in future; and the savings rate is
exogenous. And the retailers in “Division of Labor” do not seek to maximize profits, but
only to earn positive profits.
While resistance to agent-based methodology is to be expected, I hope to convince the
profession that the insights generated through agent-based simulation, especially in
“Division of Labor,” are too plausible and valuable to be ignored. More generally, while
Walrasian understanding of markets has its uses, it cannot be granted a methodological
monopoly, because its inability to deal with nonconvexities gives it an inherent,
permanent bias against the division of labor, and blinds economists to the phenomenon of
increasing returns. In Adam Smith’s vision, complexity and competition were
compatible, and jointly gave rise to economic growth. The Walrasian model creates an
artificial contradiction between complexity and competition, which prevents us from
understanding growth. That is the trap from which the models in this dissertation offer
13
ways to escape.
WORKS CITED:
Arrow, Kenneth and Gerard Debreu. 1954. “Existence of an equilibrium for a
competitive economy.” Econometrica, Vol. 22, No. 3, pp. 265-290
Axelrod, Robert; and William D. Hamilton. 1981. “The evolution of cooperation.”
Science, New Series, Vol. 211, No. 4489, pp. 1390-1396.
Lehtinen, Aki and Jaakko Kuorikoski. 2007. “Computing the Perfect Model: Why Do
Economists Shun Simulation?” Philosophy of Science, Vol. 74, pp. 304-329
Romer, David. 2006. Advanced Macroeconomics.McGraw-Hill Irwin.
Sachs, Jeffery D.; Andrew D. Mellinger; and John Luke Gallup. 1999. “Geography and
Economic Development. International Regional Science Review, Vol. 22, No. 2, pp.
179-232.
Schelling, Thomas C. 1971. “Dynamic Models of Segregation.”Journal of Mathematical
Sociology 1:143-186.
Smith, Adam. 1776. The Wealth of Nations.
Solow, Robert. 1956. “A Contribution to the Theory of Economic Growth.” The
Quarterly Journal of Economics, Vol. 70, No. 1, pp. 65-94
14
CHAPTER 1 THE INVISIBLE HAND RELOADED
NEW MICROFOUNDATIONS FOR PRICE THEORY
Abstract: Firms compete in a single-good industry, with exogenous demand, consisting of many shoppers,
each of whom buys a quantity that is a linear, decreasing function of price. Shoppers buy from the firm
whose price, plus a firm-specific transaction cost randomly generated each turn, is lowest. Firms produce
the good, set a price, and sell to shoppers on a first-come-first-serve basis until their inventories are
exhausted. Firms are not endowed with any knowledge about the shape of the demand curve. Instead, they
guess demand by running regressions of quantity against price based on their own sales histories. This is
called the empiricist strategy, and empiricist competition results when many firms in the same industry use
it. It is flexible enough to be combined with various assumptions about cost and entry/exit. When the
number of firms is exogenous, a result close to Bertrand competition emerges: most of the monopoly
markup at n=2. With (gently) falling average costs and free entry, there emerges a result long held to be
impossible: a competitive market in which the supply curve is downward-sloping.
Monopoly (n=1, where n is the number of firms competing in an industry) and perfect
competition (which is held to emerge as n→∞) are the subjects of well-respected simple
models, and consequently they often supply the microfoundations of other models. By
15
contrast, oligopoly theory, where n is a small to medium-sized number, is a mess. The
Cournot (1838) and Bertrand (1883) models lack plausibility (because of the oddness of
their assumptions) and generality (because of the specificity of their assumptions) yet
remain the best-known models of oligopoly because no later models have proven clearly
superior. Moreover, the “theory of the core,” pioneered by Edgeworth (1881), and more
recently championed by Telser (1978, 1991, 1994, 1996), shows that for many if not most
technological specifications, equilibrium simply does not exist in the oligopoly case.
Consequently, oligopolistic competition is usually assumed away when economists want
to develop well-specified and reasonably simple models of other phenomena.
This neglect of oligopoly is costly, because it requires either a rejection of free entry (if
monopoly is assumed) or else the pervasive assumption of constant average cost (AC)
technology. Moreover, some theory of oligopoly is needed even to justify the belief that
p→c as n→∞ in the constant AC case, and therefore that perfect competition is
approximately true when n is sufficiently large.
This paper presents a new approach which, with the help of computer simulations,
transcends the traditional messiness of oligopoly theory and offers a hitherto elusive
degree of clarity, rigor, realism and generality. The approach is labeled “empiricist
competition,” and it can supply new microfoundations for price theory, and in particular,
a novel justification for the widespread use of the assumption of perfect competition in
economics, as long as the assumption is understood as only an approximation. The word
empiricist is a double allusion, (a) to the British empiricist school of 18th-century
16
philosophy, and (b) to the econometric techniques of contemporary empirical economics,
which are imitated by firms in the empiricist competition model. From the British
empiricists I borrow the idea that our knowledge is derived from perception. In
empiricist competition, firms know their own cost functions, but derive all their
knowledge about market demand from experience, in particular from the prices they have
charged and the quantities they have sold in the past. Using this information, firms
behave just as a modern econometrician might: they run a regression of quantity against
price, and then maximize profits as if they were monopolists in the market they observe.
This is called the empiricist strategy. Customers buy from the seller with the lowest price
plus a random “transactions cost” specific to each customer, firm, and turn. Transactions
costs ensure that each firm typically faces a downward-sloping demand curve.
Empiricist competition is applied to three kinds of cost functions: (a) constant costs, (b)
U-shaped average costs, and (c) gradually falling average costs. According to traditional
theory, in case (b) the market has an empty core and there is no equilibrium, and case (c)
is a natural monopoly, but I will show how, under empiricist competition, a nearcompetitive market emerges and is sustained in all three cases. Standard neoclassical
price theory is partly vindicated. Empiricist competition shows that the critique of
standard price theory from the side of the “theory of the core” is less fatal than it might
appear. It also shows that because near-perfectly competitive outcomes can occur with
small n, standard price theory may be more generally applicable than economists would
be justified in regarding it if, for example, Cournot’s theory of oligopoly were true. Yet
17
empiricist competition requires a retreat from strict notions about competitive
equilibrium. The “zero profit condition” and the “law of one price” never quite hold.
Firms typically face demand curves that slope down at least slightly. And “exploratory”
random flux is ineradicable from the model because the information it provides to firms
is essential to the market’s stability.
Finally, the model yields at least one important novel result, namely, that whereas it has
long been held that a downward-sloping supply curve contradicts deep economic logic, or
at best that it is tenable only in the case of external economies of scale, empiricist
competition legitimizes the downward-sloping supply curve by showing how, after all,
it is consistent with the market being competitive in just the same sense that as (according
to the empiricist competition model) a flat supply curve is. This finding has major
ramifications—not wholly novel but elusive from the perspective of more traditional
methods—for the theory of long-run economic growth (the economy grows because the
market gets bigger), of international trade (economies of scale rather than comparative
advantage explain why openness is so beneficial), and of economic geography (people
congregate in cities because stuff can be supplied cheaper on a larger scale).
To support the robustness of the model, I propose two conjectures, which I then subject
to some testing, although proving them is too large a task to undertake here (and probably
exceeds my skill if it is possible at all). First, I conjecture that the empiricist strategy (or
some variation of it) is as good as or better than any other strategy for a firm operating in
18
the market environment which is created in these simulations, and that has the same
information as the empiricist firm uses. Second, I conjecture that the major result of the
paper—that oligopoly closely resembles perfect competition—systematically holds when
firms adopt strategies that are competitive with empiricist.
I. Background
A. Equilibrium, Chaos and Cyclicality
If we want to understand the preoccupation of economists with equilibrium, a good place
to look for answers is in the writings of Frank Knight, who brilliantly distilled and
channeled the heritage of Alfred Marshall, and whose students later comprised the core of
the Chicago School, which gained great influence in the late 20th century by championing
the neoclassical, equilibrium economics which for some decades had been partly eclipsed
by Keynesianism. In his 1921 dissertation essay Risk, Uncertainty and Profit, Knight
justifies the ubiquity of the equilibrium concept in economics with a metaphor from the
natural sciences:
The concept of equilibrium is closely related to that of static method. It is the nature of every
change in the universe known to science to have "final" results under any given conditions, and
the description of the change is incomplete if it stops short of the statement of these ultimate
tendencies. Every movement in the world is and can be clearly seen to be a progress toward an
equilibrium. Water seeks its level, air moves toward an equality of pressure, electricity toward a
uniform potential, radiation toward a uniform temperature, etc. Every change is an equalization of
the forces which produce that change, and tends to bring about a condition in which the change
19
will no longer take place. The water continues to flow, the wind to blow, etc., only because the
sun's heat—itself a similar but more long-drawn-out redistribution of energy—constantly restores
the inequalities which these movements themselves constantly destroy. (Knight, 1921, Kindle
edition)
Knight seems to be in error here, and it is instructive to point out how. It is not the case
that “every movement in the world is… a progress toward an equilibrium.” The orbits of
the moon around the earth and the earth around the sun, for example, are sustainable
cyclical movements with no tendency to slow down or cease; the same holds for electrons
orbiting the nucleus of an atom. The atoms of a warm object are not, as they seem to us,
stationary, but are in constant irregular, or chaotic, motion. Chaos and cyclicality are at
least as common in nature as equilibrium.
Yet there is also a sense in which a pattern of cyclical or chaotic motion can be regarded,
from a different, a higher or more abstract, perspective, as an equilibrium. Thus, though
the molecules of the table are in constant motion, the table itself is a stable object, an
equilibrium. This paper will suggest that the microfoundations of price theory should
take account of cyclicality and chaos; that equilibrium can still usefully describe markets
at a higher level; but that micro-level cyclical or chaotic motion may have important
effects on the nature of that equilibrium and must not be ignored.
B. Equilibrium and the Critique from Core Theory
20
There are many critiques of the neoclassical idea of equilibrium, of which the most
eloquent is that of the Austrian school, as summarized in Kirzner (1997). But more
devastating, because it accepts many neoclassical assumptions and shares its
mathematical and technical bent, is “core theory,” which begins with the well-known
Edgeworth box model, shown in Figure 1.
A
XA
Initial
endowment
YA
YA
+
Contract
curve or
“core”
XB
XA + XB
Figure 1: The "core" in an Edgeworth box model
21
YB
B
Figure 1 represents the bargaining problem for two agents, A and B, with endowments,
respectively, of (X A , Y A ) and (X B , Y B ), where X and Y are two goods. The width of the
box represents X A + X B , the total quantity of good X available, while the height of the
box represents Y A + Y B . The grey curves are A’s indifference sets, the dashed curves,
B’s indifference sets, with curves further from A’s and B’s origins representing higher
utility for A and B, respectively. Every point in the box represents a feasible allocation
of X and Y to A and B. The shaded area represents the set of allocations which are
Pareto-superior to the initial endowment, and in that sense represent possible deals that
might be struck between A and B to the advantage of both.
The black line represents the set of allocations which are (a) Pareto-superior to the
endowment, and also (b) Pareto-optimal. This set of allocations is called the “contract
curve” or “core” of the market. In the core, (a) no agent can benefit by deviating
unilaterally, and (b) no group of agents benefit by deviating as a coalition. Condition (b)
is contentless with two agents, since the only coalitions are A and B separately and the
A—B coalition, but it becomes important when n>2. (A Nash equilibrium requires only
condition (a). Accordingly, every allocation in the core is a Nash equilibrium, but not
necessarily vice versa.)
So far, so conventional; but this line of inquiry suddenly becomes subversive when it
turns out that the core may be empty, that is, there may be no allocation which is both
22
Pareto-superior to the endowment and from which no agents or coalitions benefit by
recontracting, in short, no competitive equilibrium.
The following is an example of a market with an empty core. There are (a) three agents,
with (b) positive endowments of two goods, and (c) nonconvex utility functions in this
two-dimensional goods space. For concreteness, let A, B, and C have utility functions
U i ( X i , Y=
X i 2 + Yi 2 , i=A, B, C, and endowments of one unit of X and one unit of Y.
i)
The initial endowment is not Pareto-optimal because agents are better off specializing.
Starting from the endowment, any bilateral trade creates surplus utility. The minimum
quantity of Y an agent will accept in return for a unit of X is
2 − 1 ≈ 0.41 . An agent
who is on the other end of such a deal gets the maximum utility of 8 − 2 2 ≈ 5.17 . If
two agents, say A and B, make a deal that excludes C, C can offer either of them up to
5.17 in utility (that is, offer one unit of X for anything above 0.41 Y) to renege on the
deal, and C’s offer will be superior, for at least one of the agents, to the A—B deal.
When it is accepted, say by A, there is a new outsider, B, who can make a similar offer
and break up the A—C deal. This leads to cycling, which continues without any
terminus. Nor does it help to make a three-way deal. For any A—B—C deal that can be
made, there is one or more “coalitions” of agents that will be better off by reneging and
forming a bilateral deal.
Empty cores are especially likely to arise in cases involving production. For example, if
the Bertrand duopoly model is modified so that firms face a capacity constraint, it results
23
in “Edgeworth cycles,” (Edgeworth, 1881; Noel, 2003; Maskin and Tirole, 1988)
whereby duopolists keep cutting prices until the profits available in the residual market—
the market that cannot be served by a firm’s competitor due to capacity constraints—
exceed those to be obtained by continuing the fight for market share, at which point
prices jump upwards as one firm raises its price, its rival follows, and a price war
resumes.
A similar example is the “Viner industry,” (Viner, 1932) where firms have “U-shaped
average costs” and a minimum efficient scale; in this case, too, the core is empty. But
since any industry with fixed costs and rising marginal costs is a Viner industry, the
conclusion seems unavoidable that many industries are Viner industries, and markets
with empty cores are common, if not typical.
C. The Invisible Hand
At stake here is nothing less than the validity of Adam Smith’s notion that the Invisible
Hand of competition leads to stable, optimal outcomes (Smith, 1776). If most markets in
the real world, particularly on the supply side, are full of non-convexities, then
competitive equilibrium in the strict neoclassical sense not only fails to apply in fact, but
fails to exist as an abstract possibility. A certain argument pioneered by Walras (1874)
and culminating in Arrow and Debreu (1954) in support of Adam Smith’s claims that
market outcomes are stable and optimal is thus eviscerated. Of course, as Vriend and
Kochugovindan (1998) emphasize, neoclassical equilibrium theory is only one way of
24
conceiving, or expressing, or supporting the Invisible Hand hypothesis, but if that theory
turns out to be untenable except in implausible special cases, the Invisible Hand
hypothesis is to some extent weakened, at least until an alternative justification for the
hypothesis is discovered.
It is not easy to judge the neoclassical idea of equilibrium empirically. Whether prices, in
particular, tend to be in a state of “equilibrium” is debatable. On the one hand, most
relative prices seem to maintain fairly consistent average levels over extended periods of
time, or if there are trends, reasons for the change can be identified. On the other hand,
prices on Wall Street, at the pump, or in the supermarket fluctuate within, say, a 1%, 5%,
or 20% range from week to week, for no obvious reason.
Even if the price of peaches is the same today as yesterday, that may represent
“stickiness”—frequent price adjustment is costly—rather than “equilibrium”—prices
being at the right level to equilibrate supply and demand. A detailed study of price and
inventory dynamics in a typical store would probably, at any given time, show some
prices to be unsustainably high (inventories are accumulating) and others unsustainably
low (inventories are being run down). If core theory suggests that prices might not be in
equilibrium, but might exhibit some kind of cyclical or chaotic dynamics within a narrow
range, we should hardly reject the conclusion on empirical grounds.
25
We might, however, think that for most purposes it is preferable to ignore small-scale
price instability and interpret the state of the market as in equilibrium. And here we
glimpse the usefulness of a notion of meta- or quasi-equilibrium, of a situation that does
not satisfy strict equilibrium criteria, but exhibits a rough stable predictability over
time—like the table, comprised of moving atoms yet itself stationary. However
plausible, a quasi-equilibrium of this kind is intractable for the traditional kind of
economic theory that deals in the language of equations; but it becomes quite manageable
as soon as one adopts the techniques of agent-based simulation.
D. Agent-Based Simulation and Market Competition
Agent-based simulation (ABS) is a still comparatively recent development in socialscientific techniques. It models, or simulates, social phenomena by creating large (but of
course finite) numbers of instances of one or more classes of agents, representing various
social actors (usually people, sometimes composite entities like firms or political parties)
as “objects” in a computer’s memory. Objects are computational entities that have
instance variables, things they “know” or “have,” and methods, things they “can do.”
Objects are created and then sequentially activated. They interact with one another and
give rise to macro-level emergent patterns. Instance variables differ from agent to agent,
making the agents heterogeneous. The generality of ABS results is established not by
proving theorems but by introducing randomness and varying parameters to show that the
results are robust.
26
Axtell and Epstein (1996) provide a good introduction to the principles and possibilities
of the technique. There are several papers that apply ABS to price theory, including
Arifovic (1994), Kimbrough (2007),Vila (2008), Gintis (2007), and Makowsky and Levy
(2009); however, the present paper borrows little from them and to review these would
serve little purpose. The idea of populating a model with agents that run regressions was
advocated by Sargent (1993), who suggested that theorists should “expel rational agents
from [their] model environments and replace them with ‘artificially intelligent’ agents
who behave like econometricians.”
II. Hypothesis—in Natural Language
Although natural language can rarely attain the rigor of formal mathematics, logic, or
computation, it is highly flexible, and a result that cannot be expressed in natural
language may reasonably be suspected of incoherence. Accordingly, I will first present
an account of empiricist competition in natural language before exploring the concept
more rigorously through a simulation.
Firms want to maximize profits, but they have no certain knowledge about the shape of
the demand curve. Instead, they estimate the demand curve by regressing quantity sold
against price in recent turns (placing greater weight on the most recent turns). Based on
this demand estimate, they calculate the profit-maximizing price (as if the firm were a
“monopolist”) then perturb it slightly because price variations provide a form of market
intelligence. (They also calculate a “target inventory” and produce the quantity needed to
27
achieve it, but this gets somewhat more complex since firms want to keep buffer stocks
on hand. It is easier to describe the firm’s behavior exactly when a production function
has been specified.) This is the empiricist algorithm or strategy, the way the firm makes
decisions given a certain information set. Next, we must consider what happens when a
firm of this kind operates in different market environments.
If the firm is the sole producer of its product, and demand is downward-sloping and
stable, it behaves almost exactly like a textbook monopolist. If the firm has competitors,
it does not interact with them strategically, since strictly speaking, it does not even know
that they exist. Instead, they affect its behavior through the shape of the demand curve
that the firm observes and uses as a basis for decisions.
It is useful to assume that customer behavior is subject to a form of noise that we will call
“transactions costs.” Demand is divided among many (identical) customers, and for each
customer, each turn, and each seller, a random “transactions cost” is generated which
affects the customer’s choice of seller. The result of this noise is that demand curves do
not flatten out completely, but they become much flatter as n increases. Firms respond
by lowering their prices, as if they were trying to undersell each other, and really trying to
capture the other’s market, though firms do not know that the reason some customers are
unwilling to pay the price they are offering is because they are being supplied by the
firm’s rival. Prices are driven down to a level near cost, though still slightly above it due
28
to transactions costs (and the uselessness of producing to sell at cost earning zero profit).
This near-perfect competition can emerge quite quickly.
Because the empiricist competition model includes an equilibration process, it does not
need to have a definite end-point or equilibrium state. On the contrary, a certain degree
of cyclical and/or chaotic motion will inevitably persist in empiricist competition. Partly
for this reason, empiricist competition is well-adapted to find quasi-equilibrium outcomes
in cases where a strict equilibrium does not exist, e.g., in an industry with U-shaped AC.
Even when AC are gently falling, empiricist competition can often find a competitive
quasi-equilibrium. And because price competition can be effective under falling AC
conditions, it is possible for an increase in demand to trigger larger-scale production and
lower unit costs, resulting in lower prices, even in a market that was already competitive.
If the hypothesis can be thus outlined in natural language, what is the need for fancy
simulations to support it? First, the simulations are a useful guide to the mind and the
intuition. Often what seems obvious after it is demonstrated by a simulation is opaque
beforehand. Second, simulations can provide clear, specific, and visual representations
of what seems vague, general, and ineffable when discussed only in words. Third, verbal
arguments tend to be interminable and lack criteria for deciding arguments. Simulations
can provide compelling evidence, or in a sense, proof, of claims.
III. The Model
29
We must now study the model in a more formal and systematic way. The model
represents a single industry. Demand and input (labor) supply are exogenous (so that the
model corresponds to a “partial,” not “general” equilibrium model). The entities
comprising the model may be divided into five categories: 1 a good; consumers that buy
the good; firms that produce and sell the good; strategies that firms use to make
decisions; and the economy, within which the other entities interact. The economy
sequentially activates the two types of entities, firms and consumers, which represent
actors in time. The relationships of the main entities in the model are shown in Figure 2.
Economy
Activates
Demand
Supplies labor
Firms
Firms
Activates
Good
Bank account
Activates
Activates
Sends data
Buys
products of
Activates
Laboratory
Accumulates
data so that the
researcher can
see what is
going on in the
simulation
Demand
Number of customers
Good
Carries out
production for
the firm
Strategy
Makes price and
production
decisions for the
firm
Max price
Slope
Figure 2: Structure of the simulation (simplified UML)
1
By “entities,” I do not mean to refer to classes in the computer science sense of the term. There are many
more than five classes in the simulation.
30
There is a fixed number of identical consumers, each of which has a demand curve for
the product. Critically important to the model is the algorithm by which consumers select
sellers. They do not simply buy from the seller with the lowest price. We will see how
this intuitively appealing rule for consumer behavior causes empiricist competition to
break down due to degenerate information on the part of firms. Instead, a small
“transaction cost” is randomly generated 2 every time a consumer considers buying from a
given seller, which affects the consumer’s choice of a seller, but not the quantity she
purchases. Due to transactions costs, firms in the model always enjoy at least some
market power. 3 In general, the demand curve for an individual consumer used in the
simulation=
is q s ( pmax − p ) , where s and p max are model parameters. 4
Firms produce and sell the good. All firms have the same production function, but they
are heterogeneous in other respects. One may conceive markets populated by a fixed
number of firms, or markets characterized by free entry (and exit); both types of markets
2
Specifically, the transaction cost is a uniformly distributed random variable.
Inasmuch as firms enjoy market power, the empiricist competition model is in the tradition of
“monopolistic competition” as conceived by Chamberlin (1932). However, it does not follow the lead of
Dixit and Stiglitz (1977), who develop a way to formalize monopolistic competition at the cost of the
unrealistic assumption that all goods enter the utility function symmetrically. In this “taste for variety”
model, consumers ought to buy a little bit of everything. We should see customers walking into the
supermarket and buy a tiny quantity of every single good on the shelves. In empiricist competition, firms
face downward-sloping demand, but consumers buy only from one firm if it has adequate stocks available.
4
The consumer may not be able to buy the desired quantity from the chosen seller due to a stockout
(insufficient inventory). In that case, the consumer finds another seller by the same method, calculates the
quantity he wants to buy at the new price, and, if this quantity is more than what he has already bought,
purchases the difference. This process continues until either the consumer has bought all he wants at the
offered price, or else (but this typically does not occur) all firms’ inventories have been exhausted.
3
31
will be considered. Each firm’s instance variables include its bank account; inventory;
imputed value of inventory (for calculating profits); product (i.e., the good); price (at
which all comers may purchase arbitrary quantities until the inventory is exhausted);
archives (with data on past prices, quantities sold, and profits); strategy (for pricing and
production); and capital strategy (for deciding how much to pay in dividends). A firm’s
bank account is used to pay wages and dividends, both of which disappear from the
model, and is replenished by sales. During each activation, the inventory is replenished
by new production (unless the firm idles production) and a price is set. Between
activations, customers deplete the firm’s inventory and replenish the firm’s bank
account. 5
Firms delegate their decisions about prices and production to a strategy, an object which
uses the firms information and maps it into price and production decisions. We will
discuss strategy further in Section VII, but we must introduce immediately the Empiricist
strategy which drives most of the results in this paper. The empiricist strategy is the most
sophisticated object in the simulation. It has built-in capacities to run OLS regressions,
and, when the good has U-shaped or falling AC, to execute a numerical optimization
procedure called Newton’s method to predict the profit-maximizing price and quantity
given its beliefs about the demand curve. The empiricist algorithm uses the firm’s sales
history to regress quantity against price in order to estimate the shape of the demand
5
Throughout this paper, firms use a capital strategy that involves setting a “working capital target,” the
sum of base working capital and a multiple of past sales, and distribute as dividends any funds in excess of
this target. This strategy has no particular basis in economic theory and the only defense of it (other than
that it proved convenient for running the simulation) is that since all firms use it, it may be assumed that it
does not explain the relative success of any particular type of firm (e.g., in the tournaments in Section IV).
32
curve. Having done so, it calculates the profit-maximizing quantity and price, which
usually becomes the firm’s operational decision. The empiricist algorithm is described in
detail in Computational Appendix I, and how the empiricist algorithm works in the
constant, U-shaped, and falling AC cases is displayed graphically in Figures 3, 4, and 5,
respectively. 6 Mathematical Appendices I and II show how empiricist firms solve the
profit maximization problem in the cases of U-shaped AC and falling AC, respectively.
Goods are made, bought, and sold. We will consider three types of goods: (a) constant
AC goods, with the production function Q = AL , where A is a constant, and (b) U-shaped
AC goods, with the production function, Q= A ( L − L0 ) , 0 < α < 1 , and (c) goods with
α
gently falling average cost and the production function implicitly defined by
L=
Q
. All three types of goods are produced with labor only. Productive
1 + ln ( Q + 1)
capital assets do not exist in this model, although, as we will see, capital in another sense
does. Figures 3, 4, and 5 show examples of how the empiricist strategy works under (a)
constant AC, (b) U-shaped AC, and (c) falling AC.
6
In Figures 3, 4, and 5, it is not quite visually obvious that the firm’s estimate of the demand curve is a
fitted line corresponding to an OLS regression based on the observations of P/Q shown as circular points.
Typically, the points look like they imply a much flatter demand curve than what the firm estimates. It
must be remembered, however, that the firm is regressing quantity against price, not the other way around,
and that the regression minimizes the sum of squares of horizontal distances between the points and the
regression line. Because it is traditional to put the regressand on the vertical axis and the regressor on the
horizontal axis, the regression results will often seem inaccurate; they may look more plausible readers turn
the charts on their side. Of course, this reversal of axes is a long-standing tradition in economics, which
always puts price on the vertical axis and quantity on the horizontal axis, even though quantity is usually
regarded as a function of price rather than the other way around.
33
Figure 3: How the empiricist algorithm works with constant AC
34
Figure 4: How the empiricist algorithm works with U-shaped AC
35
Figure 5: How the empiricist algorithm works with falling AC
Figures 3, 4, and 5 represent what might be called “empiricist monopoly,” situations in
which a firm using the empiricist strategy holds a monopoly in a market and tries to
maximize profits. As a visual aid, demand was made somewhat variable, otherwise price
and quantity would vary so little that the data on which the regression is based could not
well be seen. The next step is to introduce multiple firms into the model.
IV. Why Oligopoly is Competitive
When customers have different sellers to choose from and look for the lowest price, the
demand curve faced by an individual firm becomes very unlike industry demand. It is
36
typically much flatter, and its shape depends mainly on the prices of rival sellers, and the
price-sensitivity and other behavioral characteristics of customers. Rival empiricist firms
to undersell each other until the markup is driven down to levels much lower than the
monopoly case. While this result hardly comes as a surprise, the empiricist competition
model, implemented using agent-based simulation, underlines how powerful a
mechanism price competition is, even when the number of firms is small.
A. Empiricist Competition with Constant Average Costs
Figure 6 shows the price histories of two firms in empiricist competition with each other.
The production technology is constant AC, with AC=1. The simulation (after
initializing) runs for 500 turns. Only the price band between 1 and 2 is shown, but the
monopoly price would be much higher at 5.5. The maximum transactions cost is 0.5.
37
Figure 6: Duopolistic competition with constant AC has a cyclical character
It is evident from Figure 6 that, although the “law of one price” does not hold, the two
firms’ prices track each other closely. A gap between the two firms’ prices usually
exists, and it can show some short-term persistence, but it is typically less than 0.05, that
is, less than one-tenth of the maximum transactions cost shock.
Also clear from Figure 6 is that prices exhibit an asymmetric instability: they regularly
spike upward, then tend to come down slowly. These price spikes might be described as
“spontaneous collusion.” They occur when both firms, which are always subjecting their
own prices to exploratory random disturbances, happen to raise their prices at the same
38
time, and, not experiencing the usual fall in demand because their rival’s price has also
changed, conclude that they face a steeper demand curve than they had supposed, and
raise their prices. Since the maximum “empirical price shifts” are also randomized, one
firm will soon overshoot the other, experience a collapse in demand, and cut price, and
soon a pattern of price competition is re-established, which gradually pushes price back
towards AC. But when price gets close to AC, and as the demand curve becomes nearly
flat, the more vulnerable the market is to spontaneous collusion and price spikes.
Consequently, an irregular cyclicality emerges. This can be seen more clearly in Figure
7, which “zooms out” to show how the same model performs over 1,000 turns instead of
100, and by extending the vertical axis.
39
Figure 7: Cyclical peaks vary under duopolistic competition
It would hardly be worthwhile to venture a more detailed description and explanation of
this cyclicality, which is in part “artifactual,” that is, it arises from details of the
computational implementation that have no compelling economic interpretation. Some
kind of cyclicality emerges under many different model specifications, but the pattern
varies, and in general it is enough to say that stability is elusive. 7 A more important
feature of empiricist duopoly as shown in Figures 6 and 7 is that while prices fluctuate,
7
One feature of these patterns may be noted in passing: price dynamics in duopoly are heavily affected by
the parameter u, the discount factor affecting how data from past turns is weighted. A low value of u, e.g.,
u=0.5, makes prices more volatile in the short run but more stable in the long run. A high value of u, e.g.,
u=0.99, makes prices more stable in the short run, but causes the upward spikes in price to go much higher.
40
on average they are quite low, far below monopoly price P m =5.5, and not much above
P*=AC=1. Duopoly is sufficient to eliminate most of the monopoly markup.
The power of empiricist competition is exhibited even more clearly in Figure 8, which
displays 100 simulation runs, each 500 turns long, and reduced to a single point, whose
coordinates are (a) the number of firms N, set exogenously, and (b) the average price of
the good over the entire simulation run (after 500 turns of initialization). Even two firms
are sufficient to eliminate most of the monopoly markup, and by n=5 the proportion of
the monopoly markup that remains is almost negligible. 8
8
The remaining gap between P and AC is due to details of the algorithm such as how limited capital
availability and buffer stocks interact with the transactions costs shocks. It does not seem to asymptote to
zero as N increases, but it is small enough to ignore henceforward.
41
Figure 8: Price converges to costly swiftly in empiricist competition, as n increases
In addition to the simulation data, Figure 8 supplies, for comparison, (a) AC, and (b) a
curve corresponding to the results of the well-known Cournot (1838) model. There is a
qualitative similarity between the results of empiricist and Cournot competition, in the
sense that both models predict that p→c as n→∞. But empiricist competition predicts a
much faster convergence of price to cost as competition increases.
In fact, the result shown in Figure 8 bears less resemblance to the Cournot model than to
the Bertrand (1883) model, which Tirole (1990) calls a “paradox.” Bertrand’s result
seems paradoxical because since firms earn zero profits, they might as well either exit the
42
market, or try to raise prices and see what happens, even if they expect with a high
probability to get zero market share. Equilibrium behavior in the Bertrand model is
inadequately motivated. In empiricist competition, this paradox goes away, because as
competition intensifies, firms continue to earn positive, albeit slight, profits.
The result shown in Figure 8 for n>2 may be described as a “competitive quasiequilibrium.” It is not equilibrium in the strict sense, because the P=AC condition is not
met, and small-scale volatility in price and quantity persists, but it is close enough to
equilibrium that for many purposes the differences are negligible.
B. U-Shaped Average Costs
The great merit of empiricist competition vis-à-vis Bertrand, Cournot, and various gametheoretic approaches to competition is its generality. We have seen that if the production
function is characterized by fixed costs and diminishing returns, i.e., U-shaped AC,
Bertrand and Cournot are not applicable and core theory shows that no competitive
equilibrium exists, even if demand at P*=min AC is a multiple of Q* and the number of
firms is set exogenously to be exactly sufficient to satisfy demand. By contrast,
empiricist competition can readily be adapted to the U-shaped AC case, and it yields, not
a strict equilibrium, but a quasi-equilibrium similar to that observed in the constant AC
case.
43
It was shown in Figure 4 how an empiricist firm responds when faced with a downwardsloping firm demand curve and a U-shaped AC curve, and this analysis cross-applies to
the case where a rivalry between two firms makes firm demand curves more nearly flat
(but still slightly downward-sloping due to transactions costs). How such competition
plays out in the duopoly case is shown in Figure 9, designed to be closely analogous to
Figure 6. In particular, the production function
is Q 1.649 ( L − 1) , total demand
=
0.8
would =
be Q
10
(10 − P ) if all customers faced the same price, 9 and optimal quantity
9
Q*≈5, so that QD ( P= 1=
) 10= 2Q* , that is, minimum AC=1 and it is feasible for the two
firms to produce just enough to satisfy demand at that price.
9
Since “total demand” represents the sum of the individual demands of 100 different customers, who may
select different sellers and therefore face different prices, a full explanation would be more complicated.
44
Figure 9: With U-shaped AC, competitive cycles in duopoly are more irregular
If we compare Figure 9 with Figure 6, it appears that prices are a bit higher with Ushaped AC than with constant AC, as well as a bit more volatile over very short time
intervals. (Prices in the constant AC model often fell to 1.3 or so, whereas in the Ushaped AC case, they are always above 1.4.) That prices are higher with U-shaped AC is
not too surprising, since P=min AC is only technically feasible under U-shaped AC if
firms produce exactly Q*. What has certainly not changed is that empiricist competition
eliminates most of the monopoly markup even in the duopoly case. Figure 10 even
suggests that the U-shaped AC case may even exhibit less long-run volatility than
constant AC, though this depends on several parameters.
45
Figure 10: Price fluctuates within a narrow band above cost in duopoly with U-shaped AC
Figure 11, analogous to Figure 8, confirms that, with U-shaped as with constant AC, n=2
is sufficient to eliminate almost 90% of the monopoly markup; however, no further
convergence seems to occur as n→∞, and a significant markup over cost remains. Yet
the result still seems impressive enough to justify economists’ habit of assuming
“competitive markets” in situations where fixed costs and/or diminishing returns exist, so
that a strict application of core theory would condemn the assumption as illegitimate.
46
Figure 11: With U-shaped AC, more firms than n=2 do not lower the price much
In Figure 11, demand parameters are chosen for each simulation run to correspond to the
number of firms, so that the exogenously determined population of firms is exactly
sufficient to satisfy demand at P=min AC. This of course begs the question of whether
and how the market process can select the “right” number of firms in this case. A more
satisfactory account of markets with U-shaped AC requires an implementation of free
entry.
C. The Bertrand Limit
47
While the markets we have considered so far approximate perfect competition closely
enough to provide some vindication for the widespread use of the assumption in
economics, slight imperfections are built into their designs and linger in their results. The
most obvious imperfection on the input side is the assumption of exogenously-generated
transactions costs. Intuitively, we might expect that if we removed transactions costs
from its specifications, the model would perfectly replicate the abstract competitive
markets of traditional economic theory. This prediction is startlingly disconfirmed.
Zero transactions costs are easy to implement. The “volume” of transactions cost shocks
is one of the economy’s parameters, and it can be set to zero. Figure 12, which is
analogous to Figures 7 and 10, shows what happens in the duopoly case with empiricist
competition, constant AC, and zero transactions costs.
48
Figure 12: With zero transactions costs, empiricist competition cannot discipline prices
Clearly, prices are far more volatile, as well as much higher on average, than when the
volume of transactions costs shocks is 0.5. Why? The intuition that markets should
become more competitive as transactions costs fall is not completely wrong, and Figure
13, which plots transactions costs shocks and average prices across 200 simulation runs,
shows that reducing the volume of transactions costs shocks (in duopoly) does reduce
average prices, up to a point.
49
Figure 13: Price falls with transanctions costs until close to zero
Why do average prices suddenly shift upwards again when transactions costs shocks fall
below 0.1? Because the empiricist algorithm breaks down when firms do not have
enough data to work with. When customers are extremely price-sensitive, it becomes
exceedingly unlikely that the two firms’ prices will fall in a narrow enough band that they
both capture part of the market. Competition becomes winner-take-all. And the problem
with that is that firms lack statistical methods to make inferences from turns with zero
sales. Zero sales turns cannot validly be used in regressions, because clearly the demand
curve is truncated at zero. A price that resulted in zero sales might be at or above the
intercept of the demand curve with the vertical axis, and firms do not know which. Zero
50
sales turns are simply omitted from the regression, and the result of this is that when
competition has a winner-take-all character, firms only observe the industry demand
curve, along with occurrences of zero sales which they do not know how to interpret, as
well as some turns in which they capture part of the market because their rival had a
lower price but ran out of stock.
The breakdown of empiricist competition as transactions costs shocks fall to zero may
seem like a programming artifact, 10 yet it has an economically plausible interpretation.
For competition to be sustainable, markets must be imperfect enough that a firm at a
temporary price disadvantage still gets some sales. The information that firms derive
from participation in markets is a critical ingredient in their decision making, and
competition breaks down when it is too easy for firms to be pushed out of markets
altogether. A bit of noise is needed to make markets work. Though plausible, for anyone
trained in neoclassical economics, this is a counter-intuitive result.
V. Profits and Free Entry
The study of empiricist competition under U-shaped AC 11 in the last section was
incomplete, because the number of firms was set exogenously rather than emerging
through a process of free entry and exit. Since entry and exit are presumably motivated
by the search for profits (and/or avoidance of losses), it is necessary to define profit and
10
An artifact, in computer science, is a result that is accidental, typically an inadvertent by-product of
human design, which an observer is tempted to mistake for being substantive.
11
If U-shaped AC are accepted, it is a further question whether they characterize the production of a plant
or a firm, which has multiple plants. It is assumed here that the agency costs involved in organizing larger
firms are an important source of diminishing returns, so that U-shaped AC apply at the firm level.
51
capital in the context of the model. Defining “thresholds” for the profit rate above
(below) which entry (exit) occurs raises the theoretical question of whether the “zero
profit condition” should be applicable in the model, and if so, how it should be applied.
Returns on capital consist of capital gains—changes in the capital value of the firm—as
well as “dividends” that firms pay out to investors, that is, that disappear from the model
but are recorded as profits. The capital value of the firm is the value of the firm’s bank
account plus the value imputed (by the method of historic cost) to the goods in the firm’s
inventory. The firm has a “capital strategy” which determines a “working capital target”
on the basis of its recent cost experience, and then pays out the rest of the bank account in
dividends. The rate of profit from period t 0 to period t 1 is the discount rate which, if
applied to all the income received between periodst 0 and t 1 as well as to period-t 1 capital,
would make the value of the income stream and remaining principal equal to the initial
value of the capital stock. The procedure for calculating this is discussed in
Mathematical Appendix III.
The entry and exit algorithms are as follows. Each turn, with a probability p 1 , the
economy selects y firms at random, with probability proportional to capital, calculates
their profit rates over the past t turns, and introduces a new firm into the market if the
average profit rate is greater than r 1 . This is done n times. Parameters of the entry
algorithm include p 1 , y, t, r 1 , and n.
52
Next, for each firm in the economy, with one randomly chosen exception (to prevent
extinction of the industry), profits are checked, and if less than r 2 , the firm is, with
probability p 2 , placed on “probation,” or, if it is already on probation, its degree of
probation is raised. If a firm earns profits greater than r 2 at a later time, it is taken off
probation. If its degree of probation rises to three, the firm disappears. Also, firms are
removed if they are “bankrupt” or “unviable,” meaning that they do not have enough
money to cover fixed costs. Parameters of the exit algorithm include p 2 and r 2 .
A certain arbitrariness in the entry and exit algorithms is hard to avoid. Different
algorithm designs might correspond equally well to our intuitions about, or experience of,
free entry and exit, and the choice of parameters, too, is arbitrary, inasmuch as there is
little economic intuition to justify specific values of y, t, n, p 1 , p 2 , and the length of
“probation” before a firm exits. By contrast, there is plenty of economic theory to inform
our intuitions about the values the most important parameters, r 1 and r 2 , should take, for
these represent the profit thresholds that determine when firms are willing to enter, and to
stay in, the industry, and this requires a digression into capital theory.
One other point deserves mention: in Section IV, fixed costs were implemented as
avoidable costs, to ensure that firms did not lose money, since no exit procedures had
been provided for. Here, fixed costs are implemented as overhead costs that have to be
paid every turn as long as a firm is in business, whether it is producing or not. This is to
prevent firms from producing in bursts in order to take advantage of economies of scale
53
and then shutting down production while they sell off their inventories. This can be a
good strategy when demand is limited and there are avoidable costs, but it seems
unrealistic.
A. Capital Theory: A Necessary Digression
Economists and the public both use the word “profit” to refer to the income of capital, but
they mean different things by the word “capital.” Economists usually use the word
“capital” refer to productive assets, such as buildings and machines. In that sense, there
is no capital in this model. In popular language, however, “capital” is more likely to refer
to money and other financial assets like stocks and bonds. In this sense, there is capital in
the model. Should we expect, then, that capital—money—will earn profits in a
competitive market, or not?
Economists like to distinguish between (a) “normal” profit (or “accounting” profit in
competitive conditions) and (b) “economic” or “extra-normal” profit. The motivation for
this distinction is that profits tend to be competed away, so that perfect competition ought
to eliminate them completely, yet it is clear that a market economy without any profits is
not only implausible but almost inconceivable. To make room for the fact that profits are
not literally zero, profits that are no higher than the prevailing rate of profit in the
economy are interpreted as “normal” payments for the services of capital. Profits greater
than what is need to pay the market price for capital services are called “extra-normal”
and attributed to market power or imperfect competition, and it is these which
54
competition is supposed to eliminate. The “zero profit condition” means zero extranormal profits. By this account, capital earns profit because it is a productive and scarce
asset. This does not seem to apply to the empiricist competition model, in which the
production function (for simplicity) is labor-only. 12
Yet capital is present in the model in a different form: as money. It resembles capital as
conceived of by some classical economists, who thought of it as a “wage fund” which
performs the necessary task of paying workers in advance of the sale of the products that
the workers make. Since wage-fund capital is not exactly a productive asset, one might
assume that its return should be zero. But, on the other hand, wage-fund capital might
have several possible uses, and therefore a significant opportunity cost, and this suggests
that it ought to earn a positive return.
The model itself cannot answer these questions, because it deals with a single industry
within an assumed broader economy which supplies workers and customers, and now
entrepreneurs. Either approach could be easily implemented, but for the sake of brevity,
we will restrict our attention to the case where r 1 =r 2 >0, and specifically, where
r 1 =r 2 =5%.
B. Free Entry with U-shaped AC
12
Of course, it would be fairly easy to adjust the specifications of the model to allow some role for
productive capital, but it would only be worth doing if one had some idea of the kind of insight likely to
arise from this added complexity.
55
When firms require positive profits to stay in the market, the theoretical optimum in
which all firms produce at efficient scale and charge P=min AC cannot occur, since that
would require The problem is not only that the firms need a little market power to cover
their profit threshold, but also that the market does not operate nearly smoothly enough to
settle into this ideal equilibrium. Prices remain notably above min AC, and firms’
production is volatile, sales even more so, and the number of firms remains inefficiently
small, though only to a minor degree.
A detailed snapshot of a free entry economy (with U-shaped AC) is shown in Figure 14.
The specifications of the simulation are: production function, as usual,
=
Q 1.649 ( L − 1)
0.8
QD 5 (10 − p ) , with 500 customers; and entry/exit parameters p 1 =20%,
; total demand, =
n=2, y=3, t=5, r 1 =5%, r 2 =5%, p 2 =20%. 13 The vertical axis of Figure 14 shows
production. Each of the many colored series in the chart tracks the production history of
a single firm, while the straight black line represents Q*, the efficient scale of production.
13
Reminders: p1 is the probability of an agent considering entry; p2 is the probability of an agent already in
business raising its degree of probation if profits are less than r2; r1 is profit threshold for entry; r2 is the
profit threshold for exit; n is the number of firms checked in the “market research” stage of the entry
decision; t is the number of turns of past data considered during the “market research” stage of the entry
decision.
56
Figure 14: Empiricist competition does not make firms produce at efficient scale
It is clear from Figure 14 that competition is not compelling firms to produce at efficient
scale. On the one hand, firms often shut down production, without going out of business.
On the other hand, when they do produce, they usually produce quantities considerably
larger than efficient scale. One reason firms usually overshoot efficient scale is that the
productivity penalty for producing at volumes above efficient scale is relatively small. At
the same time, firms not infrequently find that they are better off producing nothing, even
though they still incur fixed costs, to avoid undermining the prices they can earn by
selling their inventories. Both entry (the appearance of new production series) and exit
(the termination of production series) are visible in Figure 14: sometimes firms run out of
57
money or find their profits are inadequate, and exit, while at other times new firms enter
the industry, lured by the profits of other firms. Firms are using the empiricist algorithm
most of the time, but sometimes lack good data and resort to rules of thumb, such as
“stockout: markup the price” and “zero sales: discount the price.”
Figure 15 shows a simulation run with identical specifications (but not the same run
shown in Figure 14), with profit rates on the vertical axis. The average rate of profit
seems most of the time to be somewhat below the threshold rate of 5%, reflecting the fact
that firms are willing to tolerate sub-threshold profit rates for some time before they
eventually exit. Average profit rates appear to be closer to 0%.
58
Figure 15: Profits vary within a wide band around the threshold profit rate
In Figure 16, price is placed on the vertical axis. In principle, the competitive price
should be 1, but it comes as no surprise by now that market competition is not efficient
enough to attain this price, since we have already seen that competition does not force
firms to produce at efficient scale. Moreover, since we have allowed firms to demand
positive profits to enter, or to stay in, the industry, price has to be above cost to give firms
a profit margin. In Figure 16, prices cluster at a level between 1.5 and 2, far below the
monopoly price of over five. Sometimes the empiricist algorithm breaks down due to
anomalies in the data, and firms go “exploring,” trying out much higher prices, but these
prices of individual firms will not affect the average price since no customer will consent
59
to pay such high prices. Firms that charge below the going price experience stockouts
and raise their prices. Price seems to be the least volatile of the series.
Figure 16: Firms’ prices track each other closely
Finally, Figure 17 puts sales on the vertical axis. The sales series for individual firms
exhibit a high degree of volatility, as the firms constantly engage in exploratory price
innovations, and small changes in the relative prices of firms can lead to big shifts in
demand from one firm to another. That the average volume of sales is well above Q*
shows, once again, that firms are operating above efficient scale.
60
Figure 17: Sales for individual firms are quite volatile, and generally above efficient scale
Now that we know a bit about what empiricist competition with free entry and U-shaped
AC looks like up close, we are ready examine it at a more aggregated level. Figure 18
tracks the number of firms (N) and the price over 1,000 turns of a simulation run.
Demand is chosen to be ten times as large, i.e.,
=
Q 50 ( L − 1) . 14 Minimum AC=1 is
0.8
shown in the Figure as a guide to interpreting the price level. Importantly, the price level
is consistently above min AC, and by a substantially margin, generally 50-60%.
Nonetheless price remains far closer to the competitive than to the monopoly price, which
14
An economy on this scale would have rendered the above charts unreadable, but for reporting aggregate
figures a large economy helps to dilute the effect of micro-level randomness and enhances the generality of
results.
61
is over five. One reason that firms do not sell at P=min AC is that there are substantially
fewer of them than the optimal number of firms N*=90. To satisfy demand, the firms
need to produce at well above efficient scale. Both N and price fluctuate, but neither any
particular form of cyclicality in either pattern, nor any correlation between the two time
series, stands out as clearly visible.
Figure 18: Free entry with profit threshold of 5% leads to prices about 50% above minimum AC
62
If we look at the pattern of profits over time in the industry, shown in Figure 19, we see
that free entry keeps these, most of the time, slightly below the threshold rate of profit
that triggers entry. Profits exhibit great short-run volatility.
Figure 19: Profits are volatile, generally below the entry threshold
The fact that firms require positive profits to stay in business explains why the number of
firms remains well below the optimum. Since firms face nearly flat demand curves, they
produce almost up to the point where P=MC. Yet a profit rate of almost 5% must prevail
63
if firms are not to exit. To reconcile P=MC with positive profits, N<<N* is necessary.
Market imperfection leads to technical inefficiency, since the same quantity of the good
could be produced at a lower cost if more firms were operating in the industry. Is this
inefficiency eliminated when the profit threshold is zero? Figure 20, based on simulation
runs similar to those shown in Figures 18 and 19 except with r 1 =r 2 =0%, shows that it
does not. While the number of firms in the industry is affected by several parameters
without an obvious economic interpretation, and especially by the exogenous rate at
which potential entrants investigate the possibilities of the market, N<N* is a pattern
observed for a wide variety of different market specifications. The intuition behind this
result is not hard to see. When there are N* firms in the industry, they can only break
even if they all produce and sell precisely Q*. The slightest disturbance of this
equilibrium makes it infeasible for N* firms to coexist in this industry.
64
Figure 20: The simulation with free entry generates fewer than optimal firms
While there is plenty of room here for further exploration of the parameter space, 15 we
will content ourselves with some general observations about free entry in markets with
15
To show in detail how market dynamics vary with the parameters of the simulation would require an
exorbitant amount of space, but a few of the author’s notes from observing much simulation data may be of
interest. The parameter y has little effect on results except when y=1. When potential entrants examine
only one incumbent firm, their limited information acts as a form of noise which causes price movements
to lose their clear cyclical pattern, but for all y≥2 the results look similar. The parameter t has striking
effects on the shape of the cycles: at t=2, N ceases to fall below about 10 to 15, whereas when t=20, cycles
are particularly smooth and pronounced, and prices are lower as well. The perhaps ironic finding, then, is
that myopia makes the market more stable but raises prices. The parameter n affects the price level in an
unsurprising way: a high n, that is, a fast rate of firm entry, keeps prices down, whereas a low n, e.g., n=1,
keeps prices higher, and also seems to disrupt the cyclical pattern of prices. The parameter p also affects
the price level: a low p, that is, low odds of incumbent firms with subnormal profits exiting, makes cycles
rather slow, while a high p speeds them up. Most of these variations in parameters leave the cyclical
pattern in price, profits, and N intact.
65
U-shaped AC. In an industry with U-shaped AC, free entry and exit can set the number
of firms exogenously to levels which, while falling short of perfect technical efficiency,
are sufficient to supply consumer demand for goods at prices not too far above marginal
cost, and far below the monopoly price. To that extent, standard price theory is again
vindicated. That results do not approximate perfect competition more closely might
reflect the rather ad hoc nature of the free entry/exit algorithms, yet it is hard to see how a
decentralized process would improve on them. Suppose we wanted to claim, for
example, that under free entry, N will immediately jump to N*. Suppose further that N*N = 2, and that three potential entrants, A, B, and C, are waiting on the sidelines to enter
the industry and compete away the remaining profit opportunities. How would A, B, and
C solve the coordination problem and ensure that only two of them enter?
C. Free Entry with Falling AC
An economist trained in neoclassical economics will not expect the above
implementation of free entry with U-shaped AC to cross-apply to the “natural monopoly”
case of falling AC. Yet it turns out that if we simply substitute the falling AC production
Variations in r1 and r2 have the strongest effect on market outcomes. Generally, the cyclical pattern appears most
clearly and consistently when r1=r2. When r1>r2, that is, when the profits entrepreneurs must anticipate in order to
enter a market are greater than those that entrepreneurs require in order to stay in a market as an incumbent, the market
can exhibit plateaus, periods during which little entry or exit occurs because the prevailing profit rate is greater than r2
but less than r1. Yet these never seem to be quite stable. If an industry is controlled by a small oligopoly, prices and
profits may creep upward until they provoke a surge of entry. When r1<r2, that is, when entrepreneurs are willing to
enter an industry for the sake of an anticipated profit rate less than that which would induce them to stay in the market
permanently, the market may, ironically, exhibit more stability precisely because of firm churn, with new firms
constantly entering as older ones retire. Whether any of these details about the relationship of parameter values to
simulation market behavior have useful interpretations, either in terms of relevance to the real world, or in terms of
application to mainstream economic theory, is not clear.
66
function L =
Q
for the U-shaped AC, we can run simulations with the same
1 + ln ( Q + 1)
free entry/exit algorithms and get results that are qualitatively fairly similar. Before
exploring why, we will examine some results.
It is necessary, however, to add one realistic modification to the empiricist algorithm with
falling AC, namely that fixed costs are introduced, and a limit is imposed on the pace of
new hiring. The labor force a firm hires in any given period cannot exceed a weighted
average of the labor force in previous turns (with last turns having the largest weight, and
a discount factor of 0.8) plus a (randomly chosen ceiling in the range of) 50% to 100%
increase. Also, a small fixed cost is imposed. Without these modifications, there would
be too strong a temptation for firms to produce in large bursts, taking advantage of
economies of scale, and then wait while their inventories run down. The surprising result
that occurs—that not natural monopoly but a form of oligopolistic competition
characterizes the industry—is not a consequence of these small modifications.
Figure 21, which is analogous to Figure 14, shows the production series over 100 turns of
the simulation. At this level of detail every simulation run looks a little different, but
Figure 21 shows the typical state of the market, with several firms competing, and the
state of the market alternating between more stable periods where all firms are engaged in
positive production, and more volatile periods in which some firms halt production or
67
exit. In its stable states the market seems to support four or five firms producing about
150 to 250 units of the good.
Figure 21: Under falling AC, firm production levels exhibit moderate volatility
Figure 22 shows the time paths of profits for various firms. As in Figure 15, the average
profit rate is just below the entry/exit threshold, and firms sometimes take outright losses.
Occasionally firms make large profits or suffer large losses, but most firms’ profits, most
of the time, remain within a fairly narrow band, from -5% to +5%.
68
Figure 22: Profits stay a bit under the threshold, with occasional large deviations
Figure 23 provides an example of what happens to price under free entry with a falling
AC technology. Price dynamics over short periods vary considerably in the model, and
while the pattern shown in the Figure should not be considered representative of all the
patterns that occur, it does show the most common pattern. Clearly, too, the market is not
in a state of equilibrium in Figure 23. Initially, a phase of price war is under way, during
which prices are steadily declining; then there is a breakdown of price discipline, after
which price competition resumes, but at an elevated level. Despite the volatility, the
market behavior shown in Figure 23 can be regarded from a larger perspective as being in
a sort of steady state, with prices fairly stable at levels not far above the minimum
feasible cost at which a social planner could satisfy market demand (shown by the black
69
line), and far below the monopoly price. The next most common pattern is that the
industry simply fails to become established at all. Very rarely, brief periods of monopoly
occur, during which prices rise to a large multiple of their competitive levels.
Figure 23: Price follows a cyclical pattern, with alternating price-cutting and shakeout phases
Finally, Figure 24 shows an example of what happens to sales with free entry under
falling AC. The sales of any given firm are somewhat volatile, as firms are always
repricing and trying to undersell each other in a tug-of-war for market share. Firms rarely
seem to be shut out of the market altogether.
70
Figure 24: Sales of individual firms are highly volatile
The most puzzling thing about the results in Figures 21 to 24 is why competition among
several firms should be sustainable at all in a falling AC industry where one large
monopolist could produce the good more cheaply than four or five rivals sharing the
market can. There are two explanations for this.
The first reason is contestable monopoly, that is, the fact that if one firm were ever to
manage to dominate the industry for a time, it would face a temptation—to which,
following the empiricist algorithm, it would soon succumb—to raise prices, and in doing
so would open up an opportunity for its competitor. In contestable monopoly models
71
(Baumol, 1982), the mere threat of entry suffices to cause a monopolist in a falling AC
industry to charge P=AC. In the simulation, a hypothetical entrant is not sufficient.
Instead, actual entry occurs, with the result that at least some of the time, natural
monopoly is replaced by rivalrous competition. If this were the only factor working
against the monopolistic tendencies of falling AC, however, it would give rise, not to
continuous competition, but to what might be called contested monopoly, in which the
industry is usually dominated by a cautious monopolist, which eventually becomes
greedy and provokes entry and a war for control of the industry, ending in the
establishment of a new (or the re-establishment of the old) monopolist. (We will not
explore this interesting scenario further, for reasons of space.)
The second reason why natural monopoly does not occur, and the reason why continuous
oligopolistic competition is (usually) sustainable, is transactions costs. Some customers
will find it convenient to buy from the smaller supplier even at a higher price. In this
sense, the argument here echoes Sraffa (1926), who claimed that there are increasing
returns in production but decreasing returns in marketing. As a firm seeks to expand its
market, it must reach beyond the most accessible customers to customers who are more
accessible to other firms. When AC aregently falling, the slight production cost
advantage enjoyed by the large producer may be insufficient to offset the marketing
advantages enjoyed by the smaller producer.
72
To see the characteristics of the falling AC industry with perfect competition, it is
necessary to observe aggregate data over a larger time period. Figure 25 shows the time
paths of average price and N, the number of firms, over a simulation run of 5,000 turns.
There are two clear patterns in Figure 25, which are not idiosyncratic but are consistently
observed across simulation runs of the same specifications and duration: (a) the number
of firms varies but is generally between about four and twelve, and (b) the market
exhibits a high degree of price discipline, with prices remaining just a little above cost,
and far below the monopoly price. The brief price spike that occurs just after turn 4,250
is the exception that proves the rule. It occurs during a moment of monopoly, when the
partly random exit process leaves only one firm in the industry. This firm suddenly
observes a much steeper demand curve, and raises its price sharply, though even at its
peak the price is below the monopoly level of more than five. High profits quickly attract
entrants, and oligopolistic competition and low prices are quickly restored.
73
Figure 25: Free entry under falling AC leads to fairly stable competition and price discipline
It is true that empiricist competition can work with falling AC only thanks to what might
be called an “imperfection” in the market, namely, transactions costs. But not only have
we shown that transactions costs are consistent with quasi-competitive outcomes in other
markets; we have seen that noise is a necessary ingredient to make any market work. So
there is nothing particularly special about markets with gently falling AC curves. But
this discovery has dramatic implications for the possible shape of the supply curve.
VI. Yes, Supply Curves Can Slope Down
74
A downward-sloping supply curve is easy to draw, but the interpretation involves hidden
difficulties. In Figure 26 (not based on simulation data), it is superficially obvious what
the equilibrium is—the equilibrium price is P*, the equilibrium quantity is Q*—but not so
easy to say what the supply curve means.
P
D
P*
S
Q*
Q
Figure 26: What does a downward-sloping supply curve mean?
A standard definition of a supply curve might be the following:
75
SUPPLY CURVE, DEFINITION A: A supply curve is a set of points representing
combinations of price and quantity such that each quantity represents the maximum
that suppliers are willing to sell for the corresponding price.
But if Definition A is an adequate definition for an upward-sloping supply curve, it does
not fit a downward-sloping supply curve, since points to the left of the curve are below it,
and suppliers are not willing to provide those quantities for those prices, while points to
the right of the curve are above it, and represent quantities suppliers should be willing to
supply. We can, then, replace the word “maximum” with “minimum” and get:
SUPPLY CURVE, DEFINITION B: A supply curve is a set of points representing
combinations of price and quantity such that each quantity represents the minimum
that suppliers are willing to sell for the corresponding price.
But there is a subtle problem here, namely, that whereas under Definition A suppliers
would be willing to sell small quantities for the going price provided Q<Q*, under
Definition B suppliers must be assured that total demand will be adequate before they are
willing to supply anything. But how can a decentralized demand side of the market
assure a decentralized supply side of the market that it will purchase at least Q*?
Behind this conundrum is a related critique, that a downward-sloping supply curve is
technically feasible only if the industry has falling AC, that is, if it is a natural monopoly,
76
but in that case, one firm will displace its rivals, and the surviving monopolist will not
behave as a price taker. A monopolist will choose both price and quantity through an
optimization process that will take into account not only the quantity demanded at any
given price, but the shape of the demand curve as a whole. The monopolist’s behavior
cannot be represented as a supply curve, that is, a single curve representing quantity as a
function of price.
In empiricist competition, firms are not strictly price-takers and enjoy some market
power; to that extent they are like the monopolist. Supply and demand jointly determine
price and quantity, as in the neoclassical model, but the supply and demand sides of the
market are not simple functions but algorithms with vectors of complex specifications.
Yet since price competition among firms causes them to behave in ways that approximate
price-taking, the idea of a supply curve still makes sense, it merely needs a slight
redefinition, such as the following.
SUPPLY CURVE, DEFINITION C: A supply curve is a set of points representing
combinations of price and quantity that emerge as outcomes from market competition
when supply-side specifications (e.g., production functions, entry/exit algorithms) are
held constant, but the scale of demand is varied (while other demand parameters are
also held constant).
77
A supply curve in the sense of Definition C might slope either upward or downward.
Causation is not assumed to run from price to quantity; rather, there may be tangled webs
of causation among the various economic variables; but we know which parameter is
causing changes in the outcomes, because only one parameter is being varied
exogenously. Specifically, we will apply variations in the slope of the individual demand
curves (not the number of customers since that confusing the issue by changing the
granularity of the demand side) to the kind of industry with free entry and falling AC that
we studied in the previous section. The results are shown in Figure 27.
Figure 27: With falling AC and free entry, a downward-sloping supply curve is possible
78
Figure 27 shows the results of five hundred simulation runs in a market for a good with
falling AC, specifically, where L = Q / (1 + ln ( Q + 1) ) . Each of the black circles
represents the average price at which the good is sold and the average quantity sold in a
given turn; the line below represents average cost. In the simulations, there is free entry,
with an entry/exit threshold of 5% profits, and maximum transactions costs are 0.5. The
maximum price is ten, while the slope is varied so that Q d at P=0, the quantity desired by
customers at a price of zero, is varied from 20 to 5,000. Although the downward-sloping
shape of the supply curve in Figure 27 is clear, it comes through still more clearly if
market scale is varied more, and the horizontal axis is shown in logarithmic scale, as in
Figure 28.
79
Figure
: Falling AC does not necessarily lead to monopoly
The downward-sloping supply curves shown in Figures 27 and 28 are not the result of
external economies of scale, which have long been held (e.g., Adams and Wheeler, 1953)
to be a necessary condition for a downward-sloping supply curve. All economies of scale
(in production) are internal to the firm.
VII. Robustness: Empiricist in a Strategic Tournament
80
The most obvious critique of empiricist competition is that the empiricist algorithm itself
lacks motivation. Even if we feel intuitively that using the empiricist algorithm is a smart
way to behave, it is not rational in the strong sense of being demonstrably the best option
for a person with full information and seeking to maximize wealth, or utility. The trouble
is that possible ways to translate a firm’s information into price and quantity decisions are
so varied that the space of such methods eludes even description, let alone the kind of
exploration and testing that might lead to a proof that the Empiricist strategy is optimal.
If the Empiricist strategy is optimal, or in that sense “rational,” we have reason to believe
(since we know by introspection that human beings are rational) that there exists a
permanent tendency for the Empiricist strategy or something close to it to become the
dominant rule of firm behavior, even if we rarely observe people or firms explicitly
behaving according to that strategy. If Empiricist is not optimal, the relevance of results
based on the Empiricist strategy may seem
Rather than attempting such a proof, I will present two conjectures about the empiricist
strategy and the empiricist competition model:
CONJECTURE 1: OPTIMALITY OF EMPIRICIST: Across a wide range of demand and
supply side conditions, the Empiricist strategy consistently performs well, and
enables firms either to match or to out-perform rivals who use other strategies. No
other strategy that uses the same information “beats” Empiricist if the contest takes
place under a sufficiently wide range of demand-side conditions.
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CONJECTURE 2: ALL STRATEGIES THAT ARE COMPETITIVE WITH EMPIRICIST LEAD
TO COMPETITIVE MARKETS: If
there are other strategies that at match or beat the
Empiricist strategy in tournaments, when these strategies are used by firms faced by
the same demand-side environment in which empiricist competition occurs, they lead
to market outcomes similar to Empiricist, with rapid convergence of p→c.
If Conjecture 1 and Conjecture 2 are both false (along with any reformulations of them
which preserve their chief significance), the results presented in Sections II to VI of this
article are of no interest. In that case, the answer to them would be that (a) real firms do
not seem to behave like that; (b) it would not be rational for them to do so; and (c) when
firms behave in ways that are more rational and/or realistic, completely different market
outcomes occur.
If Conjecture 1 or Conjecture 2 is true, on the other hand, and a fortiori if both are true,
the results of the empiricist competition model are relevant and important. If Conjecture
1 (or some reformulation of it) is true (or roughly true), firms would be rational to behave
according to the empiricist strategy, suggesting that successful firms may really use
something like it, or at least that there will be a permanent tendency for them to do so
even if many less-than-rational firms manage to survive. If Conjecture 2 is true, the
empiricist strategy can serve as a stand-in or representative of a wider range of strategies,
of which some perhaps out-perform it, but which yield similar results.
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A. Testing Conjecture 1
The method I will introduce here for testing Conjecture 1 is to conduct “tournaments”
among different strategies. The tournament consists of one hundred simulation runs, with
a number of firms randomly selected between two and twenty, of which (a) a random
number of firms between 1 and n-1 are assigned (the) Empiricist (strategy), and (b) the
rest are assigned an alternative or “challenger” strategy. The production function has
constant AC, and the transactions costs shocks volume is also varied. Average profits for
firms employing each strategy are then calculated and compared.
The first strategy presented as a challenger to Empiricist will be called Randomizer.
When using the Randomizer strategy, a firm (a) calculates a price floor based on the
production technology—with constant costs, which are assumed throughout this section,
this is simply the unit cost—then, (b) divides this by a random number between just
above zero and just below one to set the price, and (c) spends 50% of its bank account on
production, except (d) when there was a stockout in the previous turn, in which case 90%
is spent, or (e) when sales last round were less than half of present inventory, which is a
signal that the firm already has enough stocks on hand to satisfy likely demand in the
next period, and need not produce more. The oddest thing about Randomizer, of course,
is that it sets the price randomly. Since it seems clear this is not a sensible way to set
prices, we should expect Randomizer to lose to Empiricist, as is shown in Figure 29.
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Figure 29: Empiricist firms make more profits than Randomizer firms
In Figure 29, the vertical axis represents (the log of) Randomizer profits, the horizontal
axis (the log of) Empiricist profits. The 45º or “parity” line consists of points where
Empiricist and Randomizer profits are equal; everything to the left of/above this line
means more profits for Randomizers; everything to the right of/below the line, more
profits for Empiricists. That almost all the points in Figure 29 are to the right of/below
the line, some well below it, indicates that Empiricist firms consistently out-perform
Randomizers, regardless of the number of firms, the ratio of Empiricist to Randomizer
firms in the economy, and the transactions costs parameter.
84
The next strategy to challenge Empiricist may be called Exploratory. The Exploratory
strategy makes some use of the firm’s past data, though not in such a sophisticated
fashion as Empiricist. First, if there is no data in the archive, it sets prices randomly,
calling on Randomizer’s method for this. If there is past data, and the last firm was a
stockout, Empiricist raises its price by 10%, and spends 95% of its bank account on
production. If the firm experienced no sales in the last turn, it halts production and cuts
prices by 10% (or to unit cost). Otherwise, the firm sets the price to the geometric
average of the prices in the last five turns, and spends 50% of its bank account on new
production. The results of an Empiricist vs. Exploratory tournament are shown in Figure
30:
85
Figure 30: Empiricist firms generally beat Exploratory firms, too
Although unlike Randomizer, Exploratory does out-perform Empiricist, by a small
margin, in quite a few simulation runs, still is clear that in general, Empiricist dominates
Exploratory; if anything, it tends to win out by even wider margins than in the
tournament with Randomizer.
The next challenger, Price Taker, has in a sense an unfair advantage over Empiricist,
since it not only uses its own data but observes that of other firms. Its procedure is
simply to set its price to the price level, calculated as an average of all prices at which the
good was sold in the previous round, weighted by the volume of the transaction. Figure
86
31 may be interpreted as a tie—Empiricist and Price Taker are equally good strategies
that earn similar profits when placed in competition with each other.
Figure 31: Between Empiricist firms and Price Taker firms, the contest is close
Finally, we introduce a decision-making process which uses firm data thoroughly, but
which relies more on social networks than individual choice: the genetic algorithm,
described in more detail in Computational Appendix II. A firm has as ten-bit sequence of
binary variables, which tell a person how to set the price. Firms obediently accept their
strategies. Figure 32 makes it clear that for most practical situations, Empiricist and
GAStrategy are close competition for one another, but there are a few outliers in whch
GAStrategy firms dominated empiricist firms.
87
Figure 32: Empiricist vs. the Genetic Algorithm is another close contest
So far, Conjecture 1 is still standing, for two reasons: (a) Price Taker and GA Strategy
seem to perform about equally with Empiricist, not better, and (b) since Price Taker and
GA Strategy involve the firm having more information than the empiricist firm needs to
have, they cannot disprove the optimality of Empiricist, given its information. Of course,
I have explored only a tiny part of the possible strategies that could be designed to beat
Empiricist, but the method should be clear by now, and it could be pursued further by
dreaming up more strategies and seeing how they fare in tournaments against Empiricst.
B. Testing Conjecture 2
88
The way to test Conjecture 2 is simply to replace the Empiricist strategy with another
strategy that is competitive with Empiricist in tournaments, and seeing whether the
central qualitative result of the paper, that oliogopoly is competitive, continues to hold.
For example, Figure 33 shows what happens as N is varied when the GA strategy is used
by all agents in the economy.
Figure 33: With a GA algorithm, price also converges to cost fairly quickly
Monopoly and duopoly look quite different when the GA strategy prevails than when
firms are Empiricist. And N has to be a bit larger before p-c becomes very small. Yet the
89
main finding, that itdoesn’t take very many firms to achieve something near perfect
competition, seems to apply in GA world as well as in Empiricist world.
I have not offered anything like proofs of Conjectures 1 and 2. But the conjectures do at
least hold up to a least a mild amount of scrutiny.
VIII. Conclusions
What are the main takeaways from the empiricist competition model for empirical and
theoretical economists? The most general is that while economists should be as bold as
ever in applying the paradigm of perfect competition to new situations, they should be
wary of letting deductions from that paradigm constrain their reasoning about questions
that are tangentially related to perfect competition. The “zero profit condition,” for
example, is a necessary conceptual prop for perfect competition, but is quite dispensable
in the framework of the empiricist competition model, and this may have important
implications for capital theory.
“Price stickiness” is an important phenomenon for macroeconomists, and empiricist
competition promises to shed light on it. It has been assumed throughout this article that
changing prices is costless, but it would be easy to modify the model to make changing
prices costly, so that firms would only do it if they expected the resulting increment in
profits to exceed the cost of the price shift itself. An application to the effects of
90
monetary policy might involve making central bank announcements available to firms
and testing the success of strategies which take these announcements into account in
different ways. A certain degree of price stickiness is inherent in the empiricist
competition model, simply because firms only learn by experience when changes in
demand take place. An extension of the empiricist competition model to a general
equilibrium framework might shed light on macroeconomic questions, such as business
cycles.
That supply curves can slope downwards is the most important apparently testable
hypothesis derived here, though this reflects the author’s personal interests as much as it
does the inherent tendency of the empiricist competition model to shed light on this
question rather than others. Now, in a sense, there is enormous evidence in favor of the
hypothesis of downward-sloping supply curves. A downward-sloping supply curve
implies that the cost of a thing will fall as demand for it rises, and economic growth, the
great historical trend of the past couple of centuries, can almost be defined as the costs of
things falling and demand for them rising. But this pattern is usually interpreted as a
result of “technological change,” which Romer (1990) has modeled as a growth in the
stock of ideas such that at any given time, the societal production function has constant
returns to scale.
It is true that any fall in costs typically involves a change in techniques. If a producer
simply does the same thing twice as many times, he will get just twice as much output.
91
To achieve economies of scale might involve a more refined division of labor, the use of
greater capital equipment, learning-by-doing on the part of skilled workers, or some other
change in the production process that leads to higher productivity. As an example,
suppose that an automotive workshop costs $10,000 to build, after which cars can be built
for an additional $10,000 each at a rate of 10 per month. A car factory costs $5 million to
build, after which it can produce 5,000 cars per month at $5,000 apiece in addition to
overhead. The factory method is more efficient if demand for cars is large. But if the
economy only demands ten cars per month, the workshop method will be much more
efficient. If demand for cars then expands to the point where it is efficient to introduce
the factory method, this change may represent a change in technology without involving
any change in ideas, if we assume that the factory method was already known, but was
not used because of inadequate demand. (This possibility is dealt with more thoroughly,
using a different approach, in the next chapter.) It is quite common for new technologies
to need to operate at a very large scale in order to have their full effect. The internet is a
good example of this. It may be that without tens or even hundreds of millions of
participants to fuel the growth of the internet, it would have not been worthwhile to
generate the content needed to make the internet valuable to its users. So it is plausible
that downward-sloping supply curves are in fact a common feature of the economy. If
so, it is a good thing if theory can accommodate them.
92
MATHEMATICAL APPENDIX I. OPTIMIZATION WITH U-SHAPED AVERAGE
COSTS
When the production function is characterized by U-shaped AC, specifically,
=
Q A ( L − L0 ) , the firm’s problem is to choose the Q and P that (in expectation)
α
achieve their buffer stock target for the next round while maximizing “profits,” that is,
the excess of revenue (including “revenue” that the firm “pays” to itself for inventory
investment) minus cost of production. Expressed in math, the firm’s problem is:
(1)
1

α
Q


=
Π P ( Q0 + Q ) − W  L0 +  
max

 A





whereQ 0 represents the firm’s leftover inventory before it begins a new round of
production, and W represents the wage rate. However, the firm is not a price-taker;
rather, P = Pmax − ( Q + Q0 ) / S , where S is a measure of the slope of the demand curve.
Substituting this into (1), we get:
(2)
1

α
Q + Q0 
Q




Π  Pmax −
max=
 ( Q0 + Q ) − W  L0 +  
S
 A



93




The derivative of profits is:
(3)
2Q0 + 2Q W −α1 1−αα
∂Π
=−
Pmax
− A Q
∂Q
S
α
If P max >0, W>0, S>0, and α>0 (and these are all safe to assume), there must be exactly
one value of Q for which the expression in (3) will equal zero, that is, where
 ∂Π 
Q=

 ∂Q 
−1
( 0 ) , and that is the profit-maximizing quantity for the firm to produce.
However, it is not, in general, possible to solve for Q analytically. To put it a different
way, it is typically impossible to express the inverse of the derivative of the profit
function in closed form, even though this function exists, since the derivative of the profit
function is monotonically decreasing. Instead, the zeroes can be found by taking the
2 W 1 − α −α1 1−α2α
∂ 2Π
A Q
=
− −
second derivative of the profit function,
, and using
S α α
∂Q 2
Newton’s method to find
∂Π
= 0 . Newton’s method, in general—if the function is well∂Q
behaved and/or the starting point is well-chosen—finds the values of x for which a
function f(x)=0 by iteratively calculating xi +=
xi −
1
94
f ( xi )
. As i→∞, f(x)→0.
f ' ( xi )
It is important to choose a starting point that converges. It turns out that a convenient
starting place for finding maximum profits is zero profits, that is, Q such that P=AC.
That Q, however, also cannot be solved for directly. Instead, the starting point is
Q=SP max , or the quantity demanded if P=0. From there, Newton’s method is used twice.
First, Li +=
Li −
1
Π ( Li )
is iterated to find the point where Π=0, that is, where P=AC.
Π ' ( Li )
2Q0 + 2Q W −α1 1−αα
− A Q
Pmax −
α
S
is iterated to find the point where
Second, Qi +=
−
Q
1
i
−1 1− 2α
2 W 1−α α α
− −
A Q
S α α
∂Π
= 0 , that is, where
=
Q arg max Π ( Q ) .
∂Q
Q
95
MATHEMATICAL APPENDIX II: OPTIMIZATION WITH FALLING AVERAGE
COSTS
When average costs are falling, specifically when the production function is implicitly
defined by L =
Q
, the profits which it is the firm’s task to maximize are
1 + ln ( Q + 1)
defined by the equation:
(1)


Q + Q0 
Q

=
Π  Pmax −

 ( Q + Q0 ) − W 
S 

 1 + ln ( Q + 1) 
To maximize the value of (1) using Newton’s method, it is necessary to derive the
derivative of the profit rate with respect to quantity, ∂Π / ∂Q :
(2)

 Q 
W
∂Π
2
1


= Pmax − ( Q0 + Q ) −
+W 

2
S
∂Q
1 + ln ( Q + 1)
 Q + 1   (1 + ln ( Q + 1) ) 
And the second derivative:
96
(3)
 2W
2
1
WQ 
Q
1
∂ 2Π
=
−
+
−

 − 2W
2
2
2
3
2
∂Q
S (1 + ln ( Q + 1) )  Q + 1 ( Q + 1) 
( Q + 1) (1 + ln ( Q + 1) )
As before, the profit-maximizing Q is found by Newton’s method, that is, by iterating
Qi +=
Qi −
1
∂Π / ∂Q
until Q i is sufficiently close to Q i+1 .
∂ 2 Π / ∂Q 2
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MATHEMATICAL APPENDIX III: CALCULATING FIRM PROFIT RATES
If a firm’s decisions in the short run affect the excess of revenues over variable costs, the
decision vector which maximizes this excess is profit-maximizing; but to define the rate
of profit which is thus achieved, the capital value of the firm must be defined and
calculated, and the issue of time must be dealt with.
For an investor, there are two benefits to owning shares of a company: (a) dividends, and
(b) capital gains. Both of these values are measurable for firms in the simulation.
Dividends are paid out every turn. The capital value of the firm at time t is equal to the
firm’s bank account at time t plus the value imputed to the firm’s inventory, which,
following the “cost principle” in accounting (Fess and Warren, 1987, 21), is equal to the
historical production cost of inventory items. If an investor at time t will hold a stock s
for k periods, the internal rate of return is defined as the interest rate r that solves the
following equation:
=
Kt
(4)
k
∑
i =1
Dt + i
(1 + r )
i
+
Kt + k
(1 + r )
k
98
whereD t is the dividend for turn t and K t is capital at time t. Equation (3) does not lend
itself to algebraic solution but can be solved by numeric methods, a straightforward task
in a computer simulation.
99
COMPUTATIONAL APPENDIX I: DETAILS OF THE EMPIRICIST ALGORITHM
Each instance of the Empiricist class has six instance variables: u, which takes on values
of 1 or less and defines the degree to which past data is discounted in the firm’s
regression of quantity against price; stockoutMarkup, which regulates how the firm raises
its price when it has experienced a stockout; zeroSalesDiscount, which regulates how the
firm cuts its price when it experienced zero sales in the previous round;
empiricalPriceShiftFactor, which regulates the way in which the strategy user limits
price increases and decreases in response to perceived sharp shifts in demand;
priceRandomizationMax, which regulates the way strategy users shift prices at the end of
the turn for exploratory purposes; and bufferStockTarget, which regulates the quantity of
the good the firm keeps on hand in excess of what it expects to sell in order to cope with
unexpected surges in demand. The values of these variables used in most of the
simulation runs are: u=0.8; stockoutMarkup=10%; zeroSalesDiscount=10%;
empiricalPriceShiftFactor=20%; priceRandomizationMax=0.5%; and
bufferStockTarget=100%.
The Empiricist class swings into action when its activate(Firm f) method is called. This
method requires a Firm as an argument. The Firm is the entity which the strategy acts
upon. The interpretation of this, of course, is that the Firm is the entity that is
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implementing the strategy, but computationally it is convenient for the Empiricist object
itself to carry out the sophisticated calculations and manipulate the firm’s variables, for
the firm might use a different strategy and behave quite differently. Empiricist has to
make a number of checks before it can be sure whether a properly empiricist algorithm
can be executed at all. First, it checks whether the Firm has at least two turns’ worth of
data in archives, because if not, the Empiricist strategy makes no sense, and the Firm
“behaves randomly.” Second, it checks whether the Firm (a) sold anything last turn, and
(b) had leftover inventory. Only if both of these are true, that is, if the Firm had positive
sales but no stockout, can the last turn be validly interpreted as indicative of the shape of
a demand curve, since otherwise the firm’s sales were “censored” by the impossibility of
selling either a negative amount or more than the firm’s inventory. If the Firm had a
stockout last turn, it raises its price by stockoutMarkup and devotes 80% of its bank
account to production. If a Firm had zero sales last turn, with 90% probability, it cuts its
price by zeroSalesDiscount and halts production; with 10% probability, it “behaves
randomly.” Finally, if the firm had no inventory to sell last turn, it “behaves randomly.”
“Behaving randomly” consists in calculating a “unit cost” which is either the historical
average cost of the inventory the firm has on hand, or else the minimum average cost of
producing the good as reported by the good itself. (Falling AC goods just report a
random number between 0 and 0.2 as the cost “floor,” since there is no minimum average
cost to produce them.)
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The Firm then tries to estimate demand based on available data. To do so, it creates a
selection of all the turns in its archives in which sales were positive and it suffered no
stockout, and then gives each of them a weight of ut, where t is the rank of an observation
by recency; that is, the most recent observations get the most weight. It regresses
quantity (sales) against price to estimate two variables p max or the intercept of demand
with the p-axis, and the slope or ratio of the increase in quantity demanded to any
decrease in price. It is unlikely, but possible, that the demand curve will not make sense,
that is, that it will be upward-sloping because changing circumstances happen to have
caused the firm to sell more when it charged high prices in recent turns, than when it
charged low prices. In this case, the firm devotes half of its bank account to production
and varies its price up or down by up to 5%, in order to explore the demand curve.
If the firm has adequate data to estimate demand and the demand estimate makes sense,
its next step is to find out what the profit-maximizing price and quantity would be, if its
demand estimate is a true representation of the market demand it faces. How this is done
mathematically depends on the production function of the good that the Firm produces.
If it is a constant AC good, the Firm would produce half of the maximum quantity that
customers would buy at a price equal to AC. If it is a U-shaped AC good, the
optimization is solved as described in Mathematical Appendix I; if a falling AC good, as
in Mathematical Appendix II. How to deal with buffer stock presents a difficulty for this
algorithm, since the Firm needs both to accumulate and to dispose of buffer stock if it is
to deal with unexpected rises and falls in demand, but how to value buffer stock presents
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a problem familiar to accountants, who are aware that replacement cost, market value,
and historical cost may be different. Empiricist firms treated inventories as worthless for
purposes of maximizing profits, that is, they regard inventories as sunk cost and do not
attempt to project their future value and conserve it. On the other hand, they incorporate
demand for their own product (as inventory) into the demand curves they use for
optimization algorithms.
Having found out what the optimum price and quantity is, in principle, firms do not
always act on this extrapolation. In particular, firms do not, on the basis of the empiricist
algorithm, raise or lower their prices by more than a random fraction of
empiricalPriceShiftFactor. (The upper bound of the empirically-based price shift is
randomized in order to prevent “artifacts” by which different firms raise their prices at
identical rates and thus engage in spontaneous collusion because of arbitrary
programming decisions.) Also, firms with U-shaped AC or falling AC do not make large
shifts in the quantity produced on the basis of the empiricist algorithm: labor is at most
twice its average over the last five turns.
Although it has proven necessary to implement special decision-making procedures for
various special situations, as described above, at the heart of the Empiricist strategy is an
intuitively compelling procedure of estimating demand by means of an OLS regression
and then maximizing profits. In stable markets, empiricist firms use this procedure
nearly all the time.
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COMPUTATIONAL APPENDIX II: THE GENETIC ALGORITHM
The GA Strategy or genetic algorithm can be described in terms of three procedures: (a)
the apply stage, (b) the choose stage, and (c) the generate stage.
A GA strategy object contains a ten-digit bit string, i.e., series of 0s and 1s, which is
converted into a price. If the bit string digits are called x 0 , …,x 9 , the apply stage consists
of setting the price to
(5)
p=
1024
9
1 + ∑ 2i xi
i =0
wherew is the wage. (The genetic algorithm was only used in the constant AC case, so
the wage is the same as unit cost.) The GA strategy uses the same generic production
rule as Randomizer, spending 50% on production in normal times, 90% after a stockout,
0% if its inventories are sufficient to cover three times last round’s sales.
The choose stage consists in calculating the profit rate over the period during which a
strategy has been used, or the size of the archive, whichever is less, and comparing it to
an average of profit rates in the economy, weighted by capital. If the firm is earning less
profits under its current strategy than other firms are earning, it chooses a new strategy
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with 50% probability. If the firm is earning more profits than other firms, it chooses a
new strategy with 2% probability. Otherwise it continues with its current strategy.
The generate stage can involve “mutating” an agent’s current strategy or “mating” two
other strategies to get a new one. Mutation consists in flipping one randomly-chosen
digit in the bit string. Mating consists in selecting two parents from the population of
firms with probability proportional to the profit rate the firms using them are earning,
converting their prices into ten-digit bit strings, by reversing the transformation in
equation (5) and rounding the last digit to the nearest integer. (Note that even non-GA
strategies can be used as a source for “mating.”) A new ten-digit bit string is generating
by selecting one of the two parents’ digits at random.
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CHAPTER 2.THE DIVISION OF LABOR IS LIMITED BY THE EXTENT OF THE
MARKET
Abstract: As Stigler observed, there is a superficial contradiction between the two major theorems
of The Wealth of Nations, that is, (a) that price competition efficiently equilibrates supply and
demand, and (b) that “the division of labor is limited by the extent of the market.” This paper
shows that the contradiction is indeed superficial, but the reconciliation requires an escape from
the neo-Walrasian conceptual framework, since Walrasian equilibrium exists only in the special
case where all functional forms are convex, leading to interior solutions, whereas specialization
involves corner solutions. A useful alternative market model was developed by Howitt and
Clower (1999), and is re-implemented here, suitably modified so that it can be applied to the case
endogenous division of labor. Essentially, Howitt and Clower’s innovation was to create, in a
simulation setting, a decentralized retail sector of specialized traders, each of which tries to clear
the market for two goods (or for one consumption good, and money) while earning positive
profits. I show that this Howitt-Clower retail sector roughly reproduces Walrasian prices. This
done, I introduce a taste-for-variety utility function and an avoidable-cost production function.
Agents thus benefit if they can specialize and trade, but when the market is small they face
stockout and “jobout” constraints which limit their opportunity to do so. As the market grows,
either due to a larger population or to capital accumulation, agents increase their level of
specialization, and—importantly—new goods are introduced to the market, which were
technologically available when the market was small, but not economically viable to produce.
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While the model’s conclusions resemble those of Romer (1987), Yang (1991), and Becker and
Murphy (1992), it avoids symmetry among goods and other strong assumptions that make those
models hard to interpret. And it suggests an explanation of the wealth and poverty of nations,
consistent with major stylized facts, which is not available to theories that regard growth as a
consequence of the creation of nonrival ideas.
But is it possible, I wonder, to account for the large cross-country differences in growth rates that
we observe in a theoretical model that does not involve external effects of the sort I have
postulated here? -- Robert Lucas, “On the Mechanics of Economic Development,” 1988, p. 35
The hypothesis that “the division of labor is limited by the extent of the market,” which is
the title of the third chapter of The Wealth of Nations (Smith, 1776) and is henceforth
called the Smithian growth hypothesis, is consistent with many stylized facts about the
world. For example, it provides one explanation of why the world is so much wealthier
today than it was a thousand years ago, namely that there is a much finer division of
labor, both because (a) there are more people, and (b) improved transport and
communications (and institutions) allow the economic integration of people who live far
apart. It explains why there are cities, and why economic development and urbanization
tend to go hand in hand. It explains the emergence of the job as a feature of advanced
economies, that is, the detachment of production from consumption as production
becomes more specialized and consumption more varied. To put this another way, it
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explains why development tends to be a transition “from subsistence to exchange”
(Bauer, 2000). It explains why world trade has grown faster than GDP for decades, and
why economic growth has been so much faster (ceteris paribus) in open than in closed
economies. It suggests a theory of the firm (“team production”) and a theory of the
business cycle (“recalculation”).
Despite its famous pedigree, the Smithian growth hypothesis has been perennially
neglected – much less often explicitly rejected – in economic theory. The reason to deny
it, is that if “the division of labor is limited by the extent of the market” – so the argument
goes – every industry will be dominated by a single firm and/or every individual will be a
unique specialist in his field, and there would be no competitive markets. Since there
seem to be a lot of competitive markets, the Smithian growth hypothesis cannot be true,
or so Becker and Murphy (1992) argue. This depends on a fallacious reading of the
phrase “limited by.” The progress of a train’s caboose is limited by the end of the
railroad track, even though it will never reach the end of the track. This is so because it is
the nature of a train that the caboose must remain a certain distance behind the engine, so
if the end of the track marks the limit of where the engine can go, the limit of where the
caboose can go is that limit minus the length of the train. But it would be quite wrong to
conclude that the caboose is stopped for other reasons and extending the track would be
of no use. Build one more mile of track, and the caboose can go one mile further. I will
argue that because monopoly leads to special inefficiencies, e.g., opportunism and
coordination failures, agents will systematically avoid pursuing specialization to the
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limits that are technically optimal, but instead will engage in monopoly avoidance
behaviors, keeping individual production efforts inefficiently diversified, and markets
competitive. It may nonetheless be the case that if the extent of the market increases,
agents will become more specialized. Therefore, the Smithian growth hypothesis is
compatible with his other hypothesis, that the invisible hand of competition equilibrates
prices, after all. And this makes it possible to answer Lucas’s question, in the epigram, in
the affirmative, for growing division of labor can fuel economic growth without having to
postulate any special new entities, such as the external effects of Lucas (1988) or the nonrival ideas of Romer (1990).
A verbal argument can convey an intuition, but “it takes a model to beat a model.” As I
said above, the Smithian growth hypothesis is usually not explicitly denied; rather, it is
assumed away by the use, in most theoretical models, of constant returns to scale (CRS)
production technologies at the micro and/or macro level, whereas the Smithian growth
hypothesis, by contrast, implies some kind of increasing returns. The pervasive use of a
neo-Walrasian approach to modeling competition, with its requirement of convex utility
and production functions if a market-clearing price vector is (reliably) to exist, has long
made Smithian growth impossible to model, at least with fully
specifiedmicrofoundations. More recently, some clever models of Smithian growth (not
always so called) have been developed, but they all depend on a very strong assumption
of symmetry among a set of goods (e.g., intermediate goods). This allows the theorist to
model an economy where specialization is endogenous by means of what might be called
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a “representative good,” analogously to the representative agents of some
macroeconomic models. However, symmetry among goods is much more false an
assumption than symmetry among agents, since while people really do have a lot in
common, it is clear that enormous variation among goods, both in the baskets of factors
required to produce them and in the role they play in satisfying human needs, is
exogenous, fundamental, and important. The cost of the symmetry assumption is that
these models can only suggest the vaguest generalizations, and it is hardly possible to
bridge the gap between the theory and the real world so as to attempt to test them
empirically.
The approach taken in this paper is laborious and roundabout, but ultimately successful. I
begin by abandoning the traditional notion of “competitive markets.” Instead, I use a
maverick model of markets and money developed by monetary economists Peter Howitt
and Robert Clower (1998). In this model, the traditional Walrasian auctioneer is replaced
by a decentralized system of “retailers” or “specialized traders” who try to “clear the
market” for a pair of goods, for agents who approach them, while turning a small profit.
Howitt and Clower use their model to account for the emergence of money. I use the
model as a workhorse for implementing the Smithian growth hypothesis. Economists
unfamiliar with agent-based modeling—that is, most economists—may be confused by
the absence of a certain reductionist move which is habitual in the field, namely, the use
of the market-clearing assumption as a quick way to arrive at claims about prices and
allocations in a large multi-agent system. Strict equilibrium, in fact, never occurs.
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Instead, quasi- or meta-equilibrium states in which relatively small cyclical or chaotic
fluctuation persists at the micro level while major aggregate indicators are roughly stable,
or show roughly steady and sustainable trends, tend to emerge. There are no
“representative firms” or “representative agents” in the model that purport to represent,
through one analytical unit, the behavior of a multitude of persons or organizations.
Instead, each entity in the simulation is intended to represent one firm or agent. Results
are obtained not by solving a system of equations (a method that is intimately linked with
equilibrium and market-clearing assumptions which in turn require convexity and thus,
for present purposes, begs the question), but by running simulations and calculating
macro aggregates from the results. This gives the results an oddly empirical flavor, but in
fact they should be regarded as purely theoretical, albeit this is theory by intuition and
induction rather than by pure deduction. Further exploration of the properties of the
model economy could be done at slight cost compared to real world data collection, but,
on the other hand, as with other theoretical models, the results can lay claim to readers’
credence only by having elucidated the logico-mathemtical ramifications of certain
plausible, albeit stylized, assumptions, not by bringing to the table any new real-world
observations.
Section I surveys the literature most relevant to the Smithian growth hypothesis, and
critiques of the Walrasian framework which has impeded the acceptance of the Smithian
growth hypothesis by economic theorists. Section II is an exposition of the HowittClower model, and the way it is modified to serve as a workhorse for a Smithian growth
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model. Section III introduces endogenous division of labor and outlines how agents
achieve rational optimization under stockout-constrained conditions. Sections IV and V
present results, with Section IV focusing on the division of labor per se, while Section V
shows the implications for growth when capital is added to the model. Section VI
concludes.
I. Background and Literature Review
A. Smithian Growth
In a 1951 article, titled like this paper after the third chapter of The Wealth of Nations,
George Stigler made the telling observation that:
When Adam Smith advanced his famous theorem that the division of labor is limited by the extent
of the market, he created at least a superficial dilemma. If this proposition is generally applicable,
should there not be monopolies in most industries? So long as the further division of labor (by
which we may understand the further specialization of labor and machines) offers lower costs for
larger outputs, entrepreneurs will gain by combining or expanding and driving out rivals. And
here was the dilemma: Either the division of labor is limited by the extent of the market, and,
characteristically, industries are monopolized; or industries are characteristically competitive, and
the theorem is false or of little significance. Neither alternative is inviting. There were and are
plenty of important competitive industries; yet Smith’s argument that Highlanders would be more
efficient if each did not have to do his own baking and brewing also seems convincing and capable
of wide generalization (Stigler, 185).
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In formulating this dilemma, Stigler seems to be reluctant to resolve it simply by
rejecting Adam Smith’s theorem, yet that seems to be, for the most part, what the
discipline did in the decades that followed, albeit more under the influence of Solow
(1956) than Stigler (1951). The way the phrase “Smithian growth” has occasionally been
used in the literature indicates a widespread perception that it is of secondary importance.
For example, Lal (2004), in arguing for the beneficent role of empires in history, writes
that
The Abbasid empire of the Arabs linked the worlds of the Mediterranean and Indian Ocean, the
Mongol empire linked China with the Near East, the various Indian empires created a common
economic space in the subcontinent, and the expanding Chinese empire linked the economic
spaces of the Yellow River with those of the Yangtze. Finally, it was the British who for the first
time in the nineteenth century linked the whole world by maintaining a Pax enforced by their
navy… In all other cases, too… the institution of an empire-wide legal system… which [helped]
in promoting trade and commerce over a wide economic space, led to those gains from trade and
specialization emphasized by Adam Smith and thereby to Smithian intensive growth. (Lal, 37)
Lal distinguishes sharply between “Smithian” intensive growth, based on trade and
specialization, and “Promethean” intensive growth, based on new ideas and inventions:
Given limited technological progress [in most empires]… Promethean intensive growth remains a
[uniquely] European miracle… The period of Smithian intensive growth in the early stages of the
empire usually petered out. (Lal, 43)
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This reading of economic history is informed by, and reflects, the conventional wisdom
(which this paper will challenge) that long-run growth comes solely from technological
change, in the sense of new ideas and inventions—Lal’s “Promethean growth”—rather
than from specialization and trade—Smithian growth. In particular, Lal wrote at a time
when the tradition of Solow (1956), which rejected capital accumulation as a cause of
long-run growth, had been greatly reinforced by Romer (1990) and other contributions to
the theory of endogenous growth, which attributed growth to new ideas. That a sideeffect of the rise of endogenous growth theory was to redefine Smithian growth as
specifically non-technological becomes clear if we contrast Lal’s use of the phrase with a
“formal model of Smithian growth” proposed by Barkai (1969). Barkai emphasized “the
dominant role attributed [by Smith] to capital, and the strategic position of technology,
specified as an endogenous variable, in his scheme” (Barkai, 396). This paper will
support Barkai’s interpretation of Smithian growth more than Lal’s. Barkai summarizes
Smithian growth in the equation:
(1)
Y = Y  K , N , T ( t , m ) 
Where K is capital, N, labor, T, technology, t, technical efficiency of capital, and m, the
extent of the market, and Y and T are increasing functions of each of their arguments.
Since more capital leads to an increase in aggregate demand, “an increase in the stock of
capital… generates an increase of product [and] thus triggers off the technology
mechanism through the extension of the market… implying not only a move along a
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product curve… [but that] the curve itself shifts upwards at one and the same time”
(Barkai, 399).
If capital contributes to technology, we are led to Smith’s optimistic vision of (at least the
possibility of) perpetual growth as capital investment raises aggregate demand and drives
technological improvement, in contrast to the Solow model, according to which the
economic growth always ends in a steady state, but for exogenous technological change.
But Barkai’s “model” only describes the time paths of aggregate variables. He supplies
no microfoundations, and he hardly mentions the factor which Smith famously
emphasized: the division of labor.
For Kelly (1997), the phrase “Smithian growth” is used to “abstract from all other
sources of growth: private capital accumulation, technological progress, and learning by
doing,” in order to focus on “reducing costs through greater specialization… Market
expansion will be the only source of growth considered here” (Kelly, 941). Kelly’s
model is rather nonstandard, in that she endogenizes geography by treating “linkage
formation,” the infrastructural and/or social capital investment involved in a firm’s doing
business in a new area, as the only form of capital investment. The theoretical model is
the vehicle for a case study of China in the ninth to eleventh centuries, and seems to be
better suited to that purpose than it is as a contribution to general economic theory. This
is a strikingly humble role for a theory advocated with such enthusiasm by the founding
father of modern economics!
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Of course, Smith (1776) did not abstract from capital accumulation, technological
progress, and learning by doing, but Lal (2004) and Kelly (1997) have narrowed the term
to refer to what they see, in hindsight, as Smith’s most distinctive contribution, whereas
other aspects of economic growth, which Smith recognized, were better expressed by
other theorists later. If specialization and trade were quite separable from capital
accumulation and technological progress, this redefinition might be a good idea. But
Smith would not have assented to it. For him, all these elements cohere into an integrated
theory of growth, in which trade and specialization are central, but capital accumulation
expands the scope for division of labor and accelerates technological progress, while
specialization gives greater scope for learning by doing. If Smith is right about this—I
will argue that he is—then to treat Smithian growth as a separate and secondary source of
growth is to underestimate its explanatory force, and to misunderstand other elements of
growth as well.
B. The Evidence for Smithian Growth
The reason for the perennial neglect and marginalization of the Smithian growth
hypothesis is not that it is implausible or empirically disproven. The evidence,
paradoxical as it may sound, is very strong, yet difficult to quantify. It is best expressed
in the form of rambling, enthusiastic descriptions of complexity that evoke wonder, rather
than, say, regression analysis. Thus, Adam Smith’s justly famous description of the
power of the division of labor in the pin factory, and his tracing of the origins of everyday
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items to show how even “the accommodation of the most common artificer or daylaborer in a civilized and thriving country [requires] the assistance and co-operation of
many thousands” are as compelling as ever. Smith goes so far as to suggest that the
living standards of European princes did not “so much exceed that of an industrious and
frugal [European] peasant, as [those] of the latter exceed that of many an African king,
the absolute master of the lives and liberties of ten thousand naked savages.” Even if we
think Smith is exaggerating, that his account is even plausible bears witness to the
importance of the division of labor.
Smith’s method of evoking wonder through tracing the origins of everyday items is
occasionally emulated by contemporary writers. In Leonard Read’s “I, Pencil,” (Read,
1958) a lead pencil tells the marvelous narrative of its origins, and shows how “no one
single person knows how to make” a pencil and “millions of people have had a hand in
[its] creation.” That Read can say “millions” where Smith said “thousands” suggests the
greater complexity of the 20th-century economy compared to that of the 18th. Most
recently, Matt Ridley (2010) writes:
As I write this, it is nine o’clock in the morning. In the two hours since I got out of bed I have
showered in water heated by North Sea gas, shaved using an American razor running on electricity
made from British coal, eaten a slice of bread made from French wheat, spread with New Zealand
butter and Spanish marmalade, then brewed a cup of tea using leaves grown in Sri Lanka, dressed
myself in clothes of Indian cotton and Australian wool, with shoes of Chinese leather and
Malaysian rubber, and read a newspaper made from Finnish wood pulp and Chinese ink. I am now
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sitting at a desk typing on a Thai plastic keyboard (which perhaps began life in an Arab oil well) in
order to move electrons through a Korean silicon chip and some wires of Chilean copper to display
text on a computer designed and manufactured by an American firm…
More to the point, I have also consumed minuscule fractions of the productive labor of many
dozens of people. Somebody had to drill the gas well, install the plumbing, design the razor, grow
the cotton, write the software. They were all, though they did not know it, working for me. In
exchange for some fraction of my spending, each supplied me with some fraction of their work.
They gave me what I wanted just when I wanted it – as if I were the Roi Soleil, Louis XIV, at
Versailles in 1700…
Louis XIV was rich because others were poor. But what about today? Consider that you are an
average person, say a woman of 35, living in, for the sake of argument, Paris and earning the
median wage, with a working husband and two children. You are far from poor, but in relative
terms, you are immeasurably poorer than Louis was. Where he was the richest of the rich in the
world’s richest city, you have no servants, no palace, no carriage, no kingdom. As you toil home
from work on the crowded Metro, stopping at the shop on the way to buy a ready meal for four,
you might be thinking that Louis XIV’s dining arrangements were way beyond your reach. And
yet consider this. The cornucopia that greets you as you enter the supermarket dwarfs anything
that Louis XIV ever experienced (and it is probably less likely to contain salmonella). You can
buy a fresh, frozen, tinned, smoked or pre-prepared meal made with beef, chicken, pork, lamb,
fish, prawns, scallops, eggs, potatoes, beans, carrots, cabbage, aubergine, kumquats, celeriac, okra,
seven kinds of lettuce, cooked in olive, walnut, sunflower, or peanut oil and flavored with cilantro,
turmeric, basic or rosemary… You may have no chefs, but you can decide on a whim to choose
between scores of nearby bistros, or Italian, Chinese, Japanese, or Indian restaurants, in each of
which a team of skilled chefs is waiting to serve your family at less than an hour’s notice…
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You employ no tailor, but you can browse the internet and instantly order from an almost infinite
range of excellent, affordable clothes of cotton, silk, linen, wool and nylon made up for you in
factories all over Asia. You have no carriage, but you can buy a ticket which will summon the
services of a skilled pilot of a budget airline to fly you to one of hundreds of destinations that
Louis never dreamed of seeing. You have no woodcutters to bring you logs for the fire, but the
operators of gas rigs in Russia are clamoring to bring you clean central heating. You have no
wick-trimming footman, but your light switch gives you the instant and brilliant produce of
hardworking people at a grid of distant nuclear power stations. You have no runner to send
messages, but even now a repairman is climbing a mobile-phone mast somewhere in the world to
make sure it is working properly just in case you need to call that cell. You have no private
apothecary, but your local pharmacy supplies you with the handiwork of many thousands of
chemists, engineers and logistics experts. You have no government ministers, but diligent
reporters are even now standing by ready to tell you about a film star’s divorce if you will only
switch to their channel or log on to their blogs… You have far, far more… servants [than the Sun
King] at your immediate beck and call.
Do such rhapsodic passages in Smith, Read, and Ridley amount to an empirical case that
“the division of labor is limited by the extent of the market?” In one sense, yes, because
they trace the pattern of exchange so far as to suggest strongly that the web includes, if
not the whole world, then that part of the world not specially isolated from it by
geographical or political barriers—that is, it includes precisely the entire market. In
another sense, no, because it does not quantify the degree of specialization and the extent
of the market at different points in time and/or space, and show that the two quantities are
positively correlated. The very nature of the hypothesis makes that kind of empirical
confirmation very difficult, because data implies commensurability—quantities in a
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regression must have units, that is they must measure “the same” thing: dollars or years
or people or potatoes, differing perhaps in many particulars but sharing membership in
some well-defined category—whereas the whole point of Smith, Read, and Ridley is that
there is an incredible variety of different things and jobs out there. Whereas datasets
presuppose a simplified ontology and fits facts into categories so as to compare them,
Smith, Read, and Ridley exult precisely in the ontological complexity of the economy,
and the Smithian growth hypothesis implies that the categories (of jobs, or of goods)
proliferate without bound as the market grows.
Beinhocker (2006) proposes a novel measure of economic complexity: the stock keeping
unit.
Retailers have a measure, known as stock keeping units or SKUs, that is used to count the number
of types of products sold by their stores. For example, five types of blue jeans would be five
SKUs. If one inventoried all the types of products and services in the Yanomamo economy [an
Amazonian tribe], that is, the different models of stone axes, the number of types of food, and so
on, one would find that the total number of SKUs in the Yanomamo economy can probably be
measured in the several hundreds, and at the most in the thousands. The number of SKUs in the
New Yorker’s economy is not precisely known, but using a variety of data sources, I very roughly
estimate that it is on the order of 1010 (in other words, tens of billions)… The most dramatic
difference between the New Yorker and the Yanomamo economies is not their “wealth” measured
in dollars, a mere 400-fold difference, but rather the hundred-million-fold, or eight orders of
magnitude difference in the complexity and diversity of the New Yorkers’ economy versus the
Yanomamo economy. (Beinhocker, p. 9)
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In principle, empirical economists might attempt to test the Smith growth hypothesis by
counting the number of SKUs available in different economies and running regressions of
GDP against product variety. The Smith growth hypothesis would predict that, generally
speaking, the number of SKUs grows more or less pari passu with economic growth, and
that it is higher in rich countries than in poor countries and in cities than in rural areas.
For what it is worth, it seems a safe bet that these predictions would be confirmed, as
Beinhocker suggests with his comparison of New Yorkers and Amazonian tribesmen. Of
course, because a product type is a more abstract entity than a product instance (chairs is
more abstract than this chair), defining SKUs as data points would present special
difficulties. If cars are invented, at least one SKU is added to the economy. Yet since
every horse is different while mass-manufactured cars (of a particular make and model)
are the same, might one define each horse as its own SKU but all Ford Model Ts as one
SKU, and paradoxically regard the automotive revolution as a reduction of complexity,
even though it gives people a very important and desirable new transportation option?
Why not, exactly?
For the moment, Stigler’s admission that “Smith’s argument that Highlanders would be
more efficient if each did not have to do his own baking and brewing also seems
convincing and capable of wide generalization” is sufficient empirical warrant. What is
needed is not evidence that the Smithian growth hypothesis is true—there is plenty of
that, unless we think we have theoretical reasons to regard it as illusory—but a theoretical
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model that can express the Smithian growth hypothesis in a formal way that clarifies its
internal logic, its assumptions, and its ramifications. Three recent attempts deserve
particular mention.
C. New Models of the Division of Labor
The perennial neglect of Adam Smith’s ideas about the division of labor is a theme of
intellectual histories such as (for those who know how to look for it) Robert Heilbroner’s
The Worldly Philosophers and (more explicitly) David Warsh’s Knowledge and the
Wealth of Nations. The same diagnosis about the history of economic thought has been
skillfully made by two top theorists, Nicholas Kaldor (1979) and Paul Krugman (1997).
But beginning in the 1980s, the technical barriers to modeling the division of labor began
to be broken. Three papers, by Romer (1987), Becker and Murphy (1992) and Yang and
Borland (1991), are particularly good representatives of this literature.
Romer (1987).Romer follows Ethier (1982) in reinterpreting the CES 16 utility function
pioneered by Dixit and Stiglitz (1977) as a production function, and derives the
conclusion that Y ( L, {M , N } ) = M 1−α ( L1−α N α ) , where N is a measure of intermediate
inputs, L is labor, and M is the number of goods (or, we might say, of SKUs). That is, in
the world of Romer (1987), the greater the division of labor, the more productive the
economy. This points to an infinite division of labor, but Romer avoids this by
introducing a quasi-fixed cost in the production function for intermediate goods. Romer
16
Constant elasticity of substitution.
126
postulates a “primary input” Z, used in producing intermediate goods subject to the quasifixed costs, and shows that in equilibrium both M and N will be proportional to Z,
yielding
(2)
Y = AZL1−α
Romer thus formalizes the Smithian growth hypothesis: the division of labor is limited by
the extent of the market. It is worth exploring some radical ramifications of equation (2)
that Romer neglects.
First, if Z is interpreted as capital, (2) implies that, contrary to Solow (1957), capital
accumulation alone can drive growth indefinitely. It does so partly by intensifying, or
extending, the division of labor, but without requiring the development of new
knowledge. In contrast not only with Solow (1956) but also with Romer (1990), Romer
(1987) is consistent with rich countries staying ahead as they pile up capital.
Second, Romer (1987) offers an explanation of why globalization—plugging into the
world economy, with its finer division of labor—seems to be the best way for poor
countries to close the gap with rich countries, and why geographical factors that facilitate
globalization—above all, access to the sea—bless the world’s most prosperous regions
(Western Europe, the US’s eastern seaboard, the Pacific Rim of Asia) while the
landlocked Rift Valley of East Africa, Central Asia and Afghanistan, and the Andean
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region in South America, remain very poor. And it explains why urbanization seems to
be a necessary, though not sufficient, condition for economic development. Of course,
the predictive successes of Romer (1987) are simply those of the Smithian growth
hypothesis which it formalizes. Barkai remarked on the same features of Smithian
growth, and they will also emerge from the model developed in this paper.
But while Romer (1987)’s microfoundations have the merit of being tractable, they are
not plausible. The symmetry among intermediate goods, both in the way they are
produced and in the way they enter the production function, is so unlike reality that the
theory is hard to interpret. And this symmetry assumption cannot be relaxed without
spoiling the equilibrium, which, following Dixit and Stiglitz (1977), depends on
substitutable goods continuing to be introduced until each producer’s U-shaped AC curve
is (simultaneously) tangent to the downward-sloping demand curve, which is possible
only with perfect symmetry. If there are any asymmetries—if some goods are cheaper to
make, or less substitutable, than others—positive profits will be available in some
industries if there is a single monopoly seller. Positive profits will attract entry; some
industries become duopolistic or oligopolistic and probably unstable, and the equilibrium
breaks down. And without an equilibrium, a theorist like Romer who operates in the
strict neoclassical tradition simply has no basis for claiming that any outcome is likelier
than any other.
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As a theory of comparative development, Romer (1987) is arguably more promising than
Romer (1990), his much more influential paper on “endogenous technological change,”
is. Romer’s argument that knowledge is non-rival implies (or seems to) much faster
income convergence among countries than the Solow model does, provided only that rich
countries transfer (or fail to prevent the transfer of) their technologies to poor countries,
which they can do at no cost to themselves, or indeed may profit by doing if the poor
countries thus made rich are willing to pay any royalties to their benefactors. But Romer
(1990) is less vulnerable to a critique of its microfoundations since in that model,
monopolistic competition is based on invention and protected by patent laws, suggesting
that the extreme and implausible assumption that intermediate goods are perfectly
symmetric, though needed in order to represent the economy as a system of equations,
might be relaxed without destroying the model’s result.
Becker and Murphy (1992). The simplicity and elegance of Becker and Murphy (1992)’s
formalization of the Smithian growth hypothesis may exceed even that of Romer (1987),
and by interpreting specialization as individual investment in human capital—by
recognizing that the fundamental indivisibility in the production process is the
individual—they motivate their fixed cost assumptions better than Romer (1987), though
they do not develop a general equilibrium. They use a Leontief production function—
every task needs to be performed to specification for each unit produced—and endow
each individual with one unit of time which is divided between training and production.
The larger the “team” engaged in production, the narrower each worker’s specialization,
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the greater his mastery of it, and the higher the team’s productivity. Symmetry is still a
problem, despite the authors’ claim that:
Since the distribution of s does not have a natural metric, it is innocuous for our purposes to
assume that all tasks are equally difficult and have the same degree of interdependence with other
tasks.
It is not clear whether this defense applies if, as is surely the case in reality, possibilities
for specialization are “lumpy,” in the sense that there are groups of tasks to which
relevant skills cross-apply, or tasks which cannot be further subdivided. But since
Becker and Murphy manage to arrive at the Smithian growth hypothesis with fewer
assumptions about market structure, their model is less vulnerable to critiques of its
microfoundations. 17 Their conclusion that Y = AH γ n1+θ , where H is the state of
knowledge and n is the size of the team, is similar to Romer’s (though increasing returns
are all ascribed to labor rather than capital, but this is inessential). Yet Becker and
Murphy immediately proceed to reject the Smith-Young hypothesis, at least as a
generalization with wide applicability in modern economies:
Every reasonably large metropolitan area has several, often many, persons who have essentially
the same specialized skills and compete in the same market. Pediatricians in the same HMO or
psychiatrists who work out of a Psychoanalytic Institute have closely related skills and seek
17
Romer (1987) is richer than Becker and Murphy (1992) in one important way: product diversification is
an essential element of Romer’s model, whereas Becker and Murphy do not bring in this element, assuming
in effect that even as the state of knowledge advances the set of tasks remains the same and people simply
become more effective at them.
130
patients in the same geographic market. Any publisher in a major city has access to many copy
editors and translators with very similar skills. The division of labor cannot be limited mainly by
the extent of the market when many specialists provide essentially the same skills. (Becker and
Murphy, 1992, pp. 1148-49)
Ironically, what Becker and Murphy propose as the factor that limits the degree of
specialization is (not the extent of the market but) a new concept they call “coordination
costs”—but “coordination and agency costs” better expresses their meaning—including
“principal-agent conflicts, hold-up problems, and breakdowns in supply and
communication,” which “all tend to grow as the degree of specialization increases.” In
support of this they appeal to Holmstrom (1982), and in one sense they are on strong
ground, for Holmstrom (1982) was an early contribution to what has since emerged as a
flourishing subfield, the “new institutional economics,” of which two Nobel Prize
winners, Douglas North and Oliver Williamson, are among the more prominent
champions.
However, it has become increasingly clear that major themes in this literature such as
“asset specificity” (Joskow, 1988) and “opportunism” (Williamson, 1979) make sense
only when many trading relationships have a monopoly-monopsony character, at least ex
post. The hold-up problems discussed in Holmstrom (1982) may illustrate this general
feature of the new institutional economics. Hold-up problems can arise when the sum of
the marginal contribution of each of a team’s members is greater than the total product.
If, as in Becker and Murphy (1992), a team faces a Leontief production function, each
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member can try to “hold up” the team, demanding most of the team’s joint product for
himself, on the grounds that without him, the team could make nothing, and if they do not
appease him, he will quit. But this threat ceases to be credible if there is a large market
for people with skills like his, and the team can easily hire a replacement for him at the
market rate. The hold-up problem occurs only if team members are to some extent
monopolists vis-à-vis their own skill sets. What this line of inquiry implies is that
coordination costs grow, not so much as the degree of specialization increases, but rather
as the team approaches the size of the whole market and each specialist becomes a
monopolist. Becker and Murphy conclude that
(3)
y=
AH γ nθ − C ( n ) ,
Cn > 0
That is, productivity rises with team size until the marginal benefit of another team
member is overtaken by marginal coordination cost. But since a large outside market
serves to discipline opportunism and eases coordination and agency costs, these costs are
=
=
C C ( n, N ) , Cn > 0, CN < 0 , and that
best described not
by C C ( n ) , Cn > 0 , but by
makes all the difference. A larger market reduces coordination and agency costs,
allowing for larger teams, and thus the division of labor is limited by the extent of the
market, after all.
As was pointed out in the introduction, the terminus of a train track may “limit” the
progress of a train’s caboose, even if the caboose never goes as far as the terminus. In the
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same way, the division of labor may be limited by the extent of the market, without ever
becoming as extensive as the market.
Yang and Borland (1991). The argument of this paper owes much more to Yang and
Borland (1991) and subsequent papers by Yang, than to Romer (1987). To describe my
debt to them fully would almost be to rehearse the whole argument of the paper. Yang
and Borland recognize the individual as the fundamental indivisibility that limits
specialization, like Becker, but they also develop a general equilibrium model, like
Romer, and having formalized the Smithian growth hypothesis, they believe it, as Yang’s
subsequent work in particular makes clear (e.g., Yang (2001)). Like Becker, they
recognize the role of transactions costs in limiting the division of labor, but nonetheless
show one way that growing division of labor might drive long-run growth. Their
recognition of the distinction between self-production and production for exchange is an
important window from theory onto economic history.
Yet there is much to criticize. The model has some gratuitous but innocuous oddities—
that production is a function of accumulated rather than current labor, for example—and
some disturbingly strong assumptions, such as that the number of agents equals the
number of goods, but most damaging is their assumption that:
All trade is determined in a futures market that operates at t=0, although over time producers will
gain monopoly power from learning by doing, at the time at which contracts are signed, no such
monopoly power exists. (p. 466)
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They account for this strange assumption by the need to “justify a Walrasian regime with
price-taking behavior at time t=0.” It seems that we must imagine a world where young
people, prior to gaining any professional experience, sign contracts binding their working
and trading behavior for their entire lives, or perhaps even—since Yang and Borland do
not explicitly model the succession of generation, inviting the interpretation of their
agents as perpetual dynasties—binding their children indefinitely. They then abide by
the terms of these futures contracts. This rigidity seems worse than the symmetry
assumption which they share with Becker and Murphy and with Romer, because it seems
more substantive. This is certainly not the way to reconcile markets with the division of
labor.
Current growth theory.If one were to hazard a summary of the modal view in the
profession concerning economic growth and comparative development, it might be the
following: endogenous technological change, as modeled in Romer (1990) is the main
cause of growth at the economic frontier, while institutions explain why some countries
are so much richer than others. The use of two theories to explain one phenomenon—
development—depending on whether we are thinking about historical or cross-country
comparisons, reflects weaknesses in both of these theories. Because institutions remain,
despite the contributions of Douglas North, Oliver Williamson, and many others, to some
extent a “black box,” to try to explain the forward movement of the economic frontier as
a function of institutional evolution is not inviting. The new institutional economics is
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strongest in case studies, explanations of price puzzles and regularities in the organization
of firms, and so forth. Its grand theories tend to be less rigorous than in other subfields,
and are influential because they confirm a (possibly correct, but unscientific) popular
prejudice that “corruption” or “bad government” or possibly “culture” explains why some
countries are poor. Institutional explanations tend to be vague and/or to transfer the
burden of explanation to other disciplines such as law, political science, ethics, cultural
anthropology, or history. They are resorted to mainly because the appealing, and tidier,
endogenous growth theory cannot explain international income differences at all.
When it comes to understanding the advance of the technological frontier, the
endogenous growth theory of Romer (1990) and others has emerged as the new
conventional wisdom, even though, as with the new theories about Smithian growth,
strong symmetry assumptions make interpretation and testing of the theory difficult.
Thus, Gregory Clark (2007), summarizing what economists know about growth, begins
by dismissing complexity with touching naivete:
Modern economies seem on the surface to be breathtakingly complex machines whose harmonic
operation is nearly miraculous. Hundreds of thousands of different types of goods are sold in giant
temples of consumption. The production, distribution, and retailing of these products, from paper
cups to personal espresso machines, involves the integration and cooperation of thousands of
different types of specialized machines, buildings, and workers. Understanding why and how
economies grow would seem to require years of study and Ph.D.-level training. But in fact
understanding the essential nature of modern growth, and the huge intellectual puzzles it poses,
135
requires no more than basic arithmetic and elementary economic reasoning. For, although modern
economies are deeply complex machines, they have at heart a surprisingly simple structure. We
can construct a simple model of this complex economy and in that model catch all the features that
are relevant to understanding growth. The model reveals that there is one simple and decisive
factor driving modern growth. Growth is generated overwhelmingly by investments in expanding
the stock of production knowledge in societies. (Clark, locations 3485-3495)
Yet having made this grandly reductionistic claim, and after describing the fundamental
growth equation with land, labor, capital, and knowledge, and trying to quantify the
importance of land and the value of physical and human capital, Clark makes a startling
admission:
Note, however, that when we arrive at this final truth as to the nature of modern growth we have
lost all ability to empirically test its truth. It is a statement of reason and faith, not an empirical
proposition. Physical capital can be measured, as can the share of capital income in all income in
the economy. But the generalized spillovers from innovation activities are not in practice
measurable. Nor is the total amount of activity designed to improve production processes
measurable. Investments in innovation occur in all economies. But unknown factors speed and
retard this process across different epochs and different economies. (Clark, locations 3645-57)
This anticlimax encapsulates the problem with Romer (1990): its microfoundations are so
stylized that the theory is difficult to interpret. A similar objection applies to Romer
(1987), Becker (1992), and Yang and Borland (1991), all of which assume symmetry
among goods. To move forward, it is necessary to break out of the trap of equilibrium
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and symmetry in which they are all stuck. Competition and markets must be conceived
in a way that allows for realistic diversity and peculiarity in the underlying production
and utility functions. In the process, we will make the Smithian growth hypothesis
credible on a micro as well as on a macro level.
D. The Walrasian Trap
The new Smithian growth models assume symmetry among goods (among other
unrealistic assumptions) in order to close a model by solving for an equilibrium, in the
Walrasian tradition. The cost of this lack of realism is that the models are very hard to
interpret in a manner that would make them testable. The theoretical device of Walrasian
equilibrium has long been criticized, especially by post-Keynesians (associated with the
political left) and Austrians (associated with the political right), for its lack of realism.
Clower (1999) offers this biting summary of the sins against reality of the Walrasian
model:
In neo-Walrasian theory:
1.
Although there are demands and supplies, there are no markets.
2.
There is no communication between prospective trading agents. Prospective trades are
signaled only to a central “demon.”
3.
Agents generate no observable data: “Trading plans” are stored, as it were, in the random
access memory of the mediating “demon.”
4.
There are no endogenous institutions: All behavioral logistics are imposed from outside the
theory (contrived ad hoc by theorists).
137
5.
No agent announces bid or asked prices: Rates of exchange are proposed and changed only by
a demon mediator.
6.
There is no competition among agents, because agents never interact directly.
7.
No agent voluntarily holds inventories or buffer stocks.
8.
There is no money or other medium of exchange.
9.
There is no trade (contrary to [various eminent theorists], the theory does not describe
commodity transfers from one agent to another). (Clower, 1999, p. 406)
Peter Howitt (1994) is similarly dissatisfied with neo-Walrasian theory, and outlines its
weaknesses by way of calling for a new approach:
We need to do more… than tack some disequilibrium dynamics onto existing models. We need to
build a new conceptual foundation. To think clearly about how transactions are coordinated in real
economies we need a clear conceptual account of… the mechanism of exchange, one based not
upon any methodological principles of equilibrium or disequilibrium, but upon careful modeling of
the way in which business firms organize transactions in real life… I am convinced that making
the business firms that actually create and maintain markets the central actors of economic theory
and modeling their behavior according to the rules and procedures they actually follow, rather than
according to maximization principles that make sense only in rational expectations equilibrium, is
the way to understand how complex economies adjust to technological change, or to any kind of
change (Howitt, 1994, pp. 772-773).
Howitt and Clower proceeded to build the “new conceptual foundation” that Howitt
called for, and I use it as the basis for a model of Smithian growth. Yet Clower’s rather
despairing conclusion, that “received doctrine in monetary and financial theory is ad hoc
138
and intellectually incoherent—effectively a blank slate on which specialists in money and
finance will, one hopes, in future write something worthwhile and nonfraudulent,”
suggests why most economists have been reluctant to reject the Walrasian framework.
Without it, they fear they may have little to say, at least with the clarity of formal theory,
leaving economic inquiry hostage to the vagaries of verbal argument and rhetoric, a
problem of which the political polarization into which the Austrian and post-Keynesian
schools have fallen, after rejecting the formal Walrasian framework which, whatever its
faults, provides a certain discipline and a common language for mainstream economics,
may perhaps serve as an illustration. By contrast, the simulation-based approach of
which Howitt and Clower (1999) and this paper are, methodologically, as precise and
rigorous as any formal model in the Walrasian tradition.
II. From Walras to Howitt-Clower: A Truly Decentralized Market to Mediate the
Division of Labor
The motivations of Howitt and Clower (1999) in developing their model of markets, and
my motivations in re-implementing and adapting it, are quite different. Howitt and
Clower seek an alternative to the Walrasian framework because of its lack of realism in
dealing with money and exchange.
This paper studies the mechanism by which exchange activities are coordinated in a decentralized
market economy. Our topic is not amenable to conventional equilibrium theory, which assumes
that exchange plans are coordinated perfectly by an external agent (usual unspecified but
139
sometimes referred to as “the auctioneer”) with no identifiable real-world counterpart. In contrast,
we depict transactors as acting on the basis of trial and error rather than pre-reconciled calculation,
and we start by noting that in reality most transactions are coordinated by an easily identified set
of agents; namely specialist trading enterprises. Thus we buy groceries and clothes from grocery
and clothing stories; buy or rent lodgings from realtors; buy cars from dealers; acquire medical,
legal, accounting, gardening and other services through specialist sellers; borrow, invest, and
insure through financial intermediaries. (Howitt and Clower, 1999, p. 1)
By contrast, I have avoided the Walrasian framework for its lack of generality. The
Walrasian framework allows a theorist insufficient freedom to specify utility and
production functions, because for many specifications, no Walrasian equilibrium exists.
Since this is counter-intuitive I offer the following illustration.
Consider the Triangle Economy in which three agents, A, B, and C, each have baskets of
one unit of good x and one of good y, and the production function is =
u x 2 + y 2 . Let D x
and D y represent total demand for x and y, S x and S y , total supply of x and y, and p, the
quantity of x needed to buy, or buyable for, one unit of y. The task of the Walrasian
auctioneer is to set a price which will equilibrate supply and demand of x and y, clearing
the market for both. Table 1 shows all the auctioneer’s options and the market outcomes
that each of them will produce. 18
18
While the example is my own, the insight that some economies have no Walrasian equilibria is derived
from the “core theory” pioneered by Edgeworth (1881) and championed more recently by Telser (1978).
Hicks (1989) is a masterful summary of the pervasiveness of the constant returns assumption in traditional
economics, and the difficulty of escaping it.
140
Price
Dx
Sx
P<1
P>1
P=1 (four
possible
market
outcomes)
0
3p
0
1
2
3
3
0
3
2
1
0
Market for
x clears?
NO
NO
NO
NO
NO
NO
Dy
Sy
3/p
0
3
2
1
0
0
3
0
1
2
3
Market for
y clears?
NO
NO
NO
NO
NO
NO
Table 1: There is no Walrasian equilibrium in the Triangle Economy
The Triangle Economy of Table 1 is one of a large class of situations in which the
Walrasian equilibrium fails to exist because of nonconvexities, in this case the
nonconvexity in the utility function which causes agents to seek to consume one good
exclusively rather than diversifying consumption. This class includes any model with
fixed costs of production, such as are an important element in Smithian growth. Intuition
is not entirely at a loss to guess how trading might play out in the Triangle Economy.
First, some trading will occur, since there are a lot of mutually beneficial bargains
available. Second, trading will lead to corner solutions: it will continue until at least two
of the agents’ inventories are concentrated in a single good, for as long as two agents
have positive inventories of both goods, they can benefit by further trades. But, third, the
order of trading will affect the pattern of trades. Therefore, fourth, trading outcomes
cannot be deduced merely from the utility functions and endowments, but will depend on
institutions and mechanisms of exchange. It is just such a mechanism of exchange which
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Howitt and Clower (1999) provide, and in the process, they create a market model that is
more general than the Walrasian because it can handle nonconvexities.
Yet interestingly, the particularspecifications of the trading problem in the Howitt and
Clower (1999) setup do not involve nonconvexities, and in fact their problem does have a
Walrasian solution. I derive this solution in Appendix A in order to check that my
implementation of the Howitt-Clower model generates roughly Walrasian prices in the
case where the Walrasian model is applicable.
In order to avoid the problem of agent-level optimization, while still ensuring an
economy sufficiently diversified that their specialist traders have something to do, they
assume a pattern of exogenous specialization:
Because we wish to deal with trading actions rather than consumer or producer choice, we assume
initially that each transactor “likes” only commodities that other transactors are endowed with;
specifically, we suppose that each transactor receives as endowment (“makes”) just one unit per
week of one kind of commodity, say that labeled i… and each transactor is a potential consumer of
just one other commodity, say that labeled j… Because no transactor “likes” the commodity it
“makes,” a transactor can acquire a commodity it “likes” only be trading with another transactor.
(Howitt and Clower, p. 8)
For Howitt and Clower, who are interested in “the emergence of economic organization,”
that is, of things like shops and money, the oddness and lack of realism of these
142
assumptions is acceptable. I, unlike Howitt and Clower, am interested in consumer and
producer choice, which are an essential part of the Smithian growth story, but the HowittClower production and utility functions will serve as useful placeholders while the model
of the retail sector is developed and tested.
A. Retailers
Each Howitt-Clower specialist trader or retailer trades in only two goods, but they
resemble the Walrasian auctioneer in that they try to “clear,” that is, to balance supply
and demand in, their own small markets. If a retailer is trading goods x 0 and x 1 , and
quantities S 0 and S 1 will be brought to the retailer for trading, the retailer could clear the
market by charging p0 = S1 / S0 and p1 = S0 / S1 , where p 0 and p 1 represent the prices of
x 0 and x 1 , respectively, denominated in terms of the other good, since, significantly, there
is no money in the model so far. This is roughly what they do, except that since (a)
retailers need to earn a profit to stay in business, and (b) they do not know the quantity of
each good that will be brought to market in a given round, their normal policy is instead
to charge:
(4)
S0e
S1e
p0 = e (1 + α ) , p1 = e (1 + α )
S0
S1
where α is the markup of price over cost that the retailer charges on the sale of each good,
allowing the retailer to retain a surplus of both goods, so as to build up inventories and/or
143
pay profits; and S0e and S1e are the expected supplies of goods 0 and 1, respectively.
Expected supply evolves according to the formula:
(5)
Sie,t +1= φ Sie,t + (1 − φ ) Si ,t
where ϕ is a parameter between 0 and 1, e.g., ϕ=0.9, Sie,t is expected supply of good i at
time t, and Si ,t is actual supply for good i at time t. Retailers must base their price setting
on expected supply because they set prices in advance, before they know how much
customers will want to purchase. 19
Retailers must earn profits to stay in business, since (a) they pay out 10% of their assets
as dividends to their owners every turn, (b) retailers’ owners pay a cost in labor to run
them, 20 and (c) they choose with some probability to exit whenever their owners’ utility
is lower than average. However, retailers do not explicitly maximize profits. They may
be described as pursuing “positive profits,” as Alchian (1950) argues that real firms do, or
as “satisficing,” to use the word coined by Simon (1972), but they fail to meet the
rigorous standards of “rationality” to which economists are accustomed. Of course, if we
19
The time-staggered character of the Howitt-Clower model, with price-setting followed by trading
followed by price-setting, may seem odd and arbitrary. Why cannot retailers lower their prices mid-turn if
sales are proving slower than expected? Why should all agents go shopping just once per turn, rather than
some shopping more frequently, others less? The alternative to this time-staggered model, however, is for
everything to happen at once in a Walrasian system of simultaneous equations. If everything is to occur at
once, there must be one agency to coordinate it. Non-simultaneity and decentralization are two sides of the
same coin.
20
In the Howitt-Clower model, retailers pay a “psychic cost” of doing business, plus overhead costs that
are specific to each good. In my model, the opportunity cost of retailing is not being able to engage in
production full-time.
144
were to interpret as a real person the Walrasian auctioneer, who controls all trade yet
keeps no surplus for himself at all, he would fall even further short of the standard of selfinterested rationality. So there are at least two ways to interpret the bounded rationality
of Howitt-Clower retailers. It may be seen as a realistic borrowing from evolutionary
economists like Alchian and Simon. Or the Howitt-Clower retail sector may be seen as a
placeholder for a theory that would introduce some form of profit maximization into
retailer behavior, without requiring them to have more information than it is feasible or
plausible to supply them with. 21
Equations (4) and (5), applied by a retailer in a model where demand is constant,
comprise an algorithm for finding a set of equilibrium prices, such that for each good,
supply equals demand plus a surplus payable (if inventories are in equilibrium) as profits
to the retailer’s owner. The trouble with this algorithm is that the path to equilibrium
would sometimes involve periods in which the inventory of one of the goods is negative.
That is, if the relative price of one of the goods started at too low a level, the price would
eventually converge to equilibrium, but in the meantime, one of the goods’ prices would
be too low, and the retailer would find that the excess of demand for that good over
supply would exhaust his inventory and—if allowed to do so—drive it into negative
territory. In Figure 1, which shows the time path of inventories during a process of price
adjustment, both the gradual convergence of prices to a level consistent with stable
21
The model of multi-firm competition in which firms use regression analysis to guess optimal price and
quantity behavior in an information-poor environment, described in Chapter 1, might be adapted as a rule
for retailer behavior in the Howitt-Clower model.
145
inventories, and the temporarily negative inventory of one good experienced by the
retailer, may be seen.
200
Inventory Qty
150
Inventory of Good 1
100
Inventory of Good 2
50
0
-50
1
11
21
31
41
51
61
71
81
91 101 111 121 131 141 151 161 171 181 191 201
-100
Turns
Figure 34: Without a zero bound, inventories may turn negative while a retailer finds its way to
equilibrium
It should be noted that a very high value of the parameter ϕ was chosen forFigure 34, so
that the adjustment, happening slowly, would be more visible. But the problem of
negative inventories applies equally to the case where the adjustment of (expected supply
and) prices is faster. Howitt and Clower accept this result and allow owners of retailers
to engage in “negative consumption,” but this is so counter-intuitive that I decided to
make things simpler to understand (and more realistic) by requiring retailers to have
nonnegative inventories of both the traded goods at all times. This, however, requires
retailers to have some way of dealing with stockouts.
146
It is easy enough to instruct retailers to stop selling good 1 when they run out of good 2
with which to pay for it, and vice versa. The problem is that the supply the retailer
misses out on by lacking goods to pay for it deprives the retailer of the information it
needs to seek the equilibrium price. When my stockout in good 1 prevents agents from
coming to me with supplies of good 2 to trade, I no longer acquire information about how
much of good 2 would be available at the price I have set, if I had sufficient stocks of
good 1 to buy it at that price. If I interpret the quantity of good 2 that I was able to
purchase as the supply of good 2 available at my offer price, that is, as S 2 , and if I
proceed to update expected supply on that basis, I will be locked into pricing behavior
that involves constant stockouts, and—more importantly—misses out on easy profits that
could be had by simply raising my price for good 1.
To avoid this trap, retailers trading in goods i and j that experience stockouts of good i,
though they continue to interpret payments in good j as supply of good j for purposes of
updating S ej,t , also keep track of the number of consecutive turns η i in which they have
experienced stockouts of good i, and modify the price-setting formula in (4) to:
(6)
pi =
(1 + β )
ηi
(1 + γ )
λj
S ~ei
(1 + α )
Sie
In equation (6), γ is a parameter λ j is the number of turns in which zero of the other
good,j, has been brought to market to buy good i. Zero sales is the opposite of a
147
stockout: too high a price leads to a loss of information. As long as the retailer has
enough in inventory to meet customer demand for both goods, and as long as the price of
good i is low enough that at least some of good j will be brought to market to buy it, η i
and λ j will be zero and equation (6) reduces to equation (4). But if the retailer
experiences stockouts, or zero sales, of one or both goods, it will raise one or both prices,
and if stockouts, or zero sales, persist, it will raise or lower them by continually
increasing margins until it finds a set of prices where its inventories are sufficient to meet
demand. Figure 35shows the time path of inventories during a price adaptation affected
by stockouts.
Inventory Qty
60
Inventory of good 1
50
Inventory of good 2
40
30
20
10
0
1
11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201
Turns
Figure 35: The stockout pricing policy keeps inventories above zero and eventually equilibrates
148
Figure 35shows a retailer’s inventories of two goods, as it operates in a market and seeks
gradually for the market clearing equilibrium. In the early part of this simulation, the
behavior of prices is a bit jerky, as the retailer repeatedly raises prices in response to
stockouts and then cuts them when it manages to avoid a stockout, which leads to a new
low price, and more stockouts and price hikes. Meanwhile, however, expected supply
evolves in ways that eventually render this pattern obsolete, as the firm closes in on a set
of prices that clear the market, with sufficient inventories to pay dividends just offsetting
surpluses from trade. And this is accomplished without any inventory ever falling below
zero.
Entry and exit.Each agent who does not own (run) a retailer considers establishing one
each round. The first step in his decision is to check what the average utility is of
comparable agents in the economy who are engaged in retail. If there are no retailers in
the economy, the agent initiates a retail operation with some probability p 1 . If the
average utility of comparable retailers is more than twice his own, the agent will (make a
tentative decision to) start a retailer with some probability p 2 . If it is less than twice his
own, the agent will (tentatively) start a retailer with some probability p 3 . The
probabilities p 1 , p 2 , and p 3 are important model parameters which it proved useful to set
at different levels for different simulations. Entry must be probabilistic because no
equilibrium could be attained if excess profits caused all non-retailers to open shops at
the same time. In the Howitt-Clower barter case we will consider first, each entrant
trades the good he makes and one other good, selected randomly. Later, in the Howitt-
149
Clower money case, the agent trades the good he makes and money, and in the
endogenous division of labor economy in Section III, the agent selects both goods
randomly. Initial prices are set based on market conditions. 22
The exit decision depends on a comparison of the retailer’s owner’s utility with the
average utility of a comparison group. If a retailer is better off than comparable agents
not engaged in retail, he stays in business. If not, he exits with a probability
(
)
pexit= 25% × 1 − uretailer / avg ( ucomparators ) , that is, he stays in business with probability
75% plus 25% times the share of comparable agents’ utility that he has been able to earn
in retail. This rule is motivated by the considerations that (a) there is no reason to exit if
a retailer is better off in business, (b) a retailer who has enjoyed success for a while
probably would not exit after one bad turn, and (c) the worse off he is in retail, the more
likely he will be to exit.
Profit. As mentioned above, the retailer transfers 10% of the goods in its inventory to its
owner every turn, as “dividends.” It is up to the owner, as a private agent, to transform
these “profits” into the good that is in his utility function. His success in doing so—
meaning both whether the appropriate trading opportunities are available, and what prices
he can get on them—affects his utility, which in turn affects his likelihood of staying in
22
More specifically, in the Howitt-Clower barter case, the agent searches for trading paths, direct or
indirect, between his two goods, and calculates a relative price based on the prices that apply along those
trading paths. In the money economies, both Howitt-Clower and endogenous division of labor, he
calculates the average money price of the other (non-money) good he is trading. If no market exists for the
retailer’s good, he sets the initial relative price to 1. In all cases, a relative price is determined first, after
which p1 and p2 are adjusted to yield the appropriate margins.
150
business.
Margins. In Howitt and Clower (1999), retailers face overhead costs—sunk costs of
doing business, incurred each turn—specific to the goods they trade in, as well as a
“psychic” cost of engaging in retail, which determines the profits retailers need to stay in
business, and because this profit goal is independent of scale, the agent’s markup falls
automatically as the scale of his operations rises. I found it simpler to make the margin a
parameter, α in equation (5), which is set randomly at the outset. 23 “Evolutionary” forces
then come into play. If margins are too small, a retailer cannot earn enough to stay in
business; if too large, it may lose its markets to the competition. The prevailing margin
may tend to be above or below the median value of the probability distribution function
of α, depending on competitive conditions.
B. A Barter Economy
Following Howitt and Clower (1999), we will first consider a barter economy, that is, an
economy in which no good has been established as money. Each agent’s utility function
is U ( x1 ,..., xn ) = x j , where i≠j. The agent’s problem is to trade x i for x j , directly or
indirectly, and at the best possible price. All goods are “perishable”: the agent’s
inventory is set to zero at the end of every turn. Agents know the identities of a certain
number of traders and what “trade offers” they offer, where a trade offer consists of a pay
good, a sell good, and a price. Thus, the retailers described in the last section offer two
23
Specifically, a random number x  [ 0,1] is generated, and the margin is set to x3.
151
trade offers, consisting of the same two goods but in opposite directions and at different
prices.
It was stressed above that retailers’ behave is not demonstrably “rational” in the sense of
maximizing any well-defined objective function. By contrast, strictly optimal solutions
to all the agents’ problems in this paper will exist and will be implemented (albeit agents
have only instantaneous, not intertemporal, utility functions, with money entering into the
immediate objective function like a consumption good, though it should be interpreted as
savings for the future). The optimization procedure for the Howitt-Clower barter case is
exceedingly intricate and interesting, but it has been relegated to Appendix B, since the
barter economy with an exogenous division of labor is, for us, only a stepping stone to a
monetary economy with an endogenous division of labor. The optimization process
consists of two parts: (a) “arbitrage,” the identification and exploitation of profitable
trade cycles, and (b) finding and exploiting the best “goal path” or series of trades
converting the good the agent makes to the good the agent likes. An important element
of the model is the way the agent’s memory is assumed to function. The agent’s
maximum memory capacity is a model parameter, set to ten for the results shown below.
Importantly, the complexity of the agent’s optimization problem under barter is roughly
proportional to the factorial of the maximum memory capacity; consequently, the barter
economy becomes exorbitantly expensive in computational terms when agents can
remember even twenty traders. Every turn, agents purge from memory traders who have
gone out of business, then if necessary remove more traders at random, until the number
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of traders left in memory is five less than the maximum. After that, the “economy” 24
supplies each agent with five new traders or whatever number is sufficient to exhaust its
memory capacity.
Howitt and Clower (1999) find that a “money” system—a trade pattern in which all
transactions involve, on one side, the same good, called money—is self-organizing in
their simulated economy, but I do not observe it, probably because Howitt and Clower
rely on differences in good-specific transactions costs to help the economy select a
money, and I omit this feature from the model. Also, the fact that Howitt and Clower’s
agents know only two traders drives them more forcefully to the adoption of money,
whereas I allow for longer chains of trades and therefore make money less strictly
necessary, although it is still very useful, as we will see. The distinction between a barter
economy and a money economy is illustrated graphically inFigure 36. In a barter
economy, the graph of trading opportunities is decentralized. All sorts of goods may be
directly traded with each other, yet because there are too many potential bilateral trades
for the market to mediate (for n goods, there would be n(n-1) possible bilateral trades),
for most pairs of goods a “double coincidence of wants” will be absent, and trade can be
mediated only through chains of trades. In a money economy, most or all transactions
involve a single good, “money,” and all trades can be executed in only two steps. Karl
Marx’s phrase “the cash nexus” is an apt description, for cash is precisely a nexus, the
24
The economy is a computational object at the center of the simulation which coordinates the other
objects. It does not set prices and coordinate trades, like the Walrasian auctioneer, but it does sequentially
activate agents and provides information. It may be thought of as the analog of a physical market in which
traders meet, or as the loose web of conversations by which people find out what is going on in an
economy.
153
hub of a network, which connects every good to every other good (or in another, less
evocative traditional phrase, “a medium of exchange”). 25
Barter economy
(decentralized graph)
Money economy with a “cash
nexus” (centralized graph)
Good
Trade offer
Figure 36: Barter versus money
To the extent that Walrasian equilibrium has long been regarded as a useful abstraction, it
would be reassuring if the results that emerge from the operations of a Howitt-Clower
retail sector confirmed some of the stylized predictions of the Walrasian model. In that
case, we might regard the Howitt-Clower retail sector as a candidate for filling the
25
The Walrasian economy is, in principle, a barter economy: markets clear through a centralized auction
process, goods are traded for goods, money is absent. It leaves money without a role, and schools of
economic theory that emphasize the role of money—Austrian and post-Keynesian macroeconomics, for
example—tend to veer off into heterodoxy, because the Walrasian model at the heart of neoclassical
orthodoxy cannot deal with their concerns.
154
Walrasian “black box.” Specifically, we might hope to find (a) that the Howitt-Clower
retail sector is able to bring the economy into some kind of equilibrium or nearequilibrium, (b) that the equilibrium is efficient or close to efficient, and (c) that the
relative prices which a Howitt-Clower retail sector establishes among goods are similar to
those which a Walrasian auctioneer establishes.
Equilibrium.The Walrasian model is atemporal, and “equilibrium” means a state of
balance, such that there will be no pressure for change over time. The Howitt-Clower
model has an explicit time aspect and is in perpetual motion at the micro level, but it
generally exhibits macro-level near-stability which can be interpreted as “equilibrium.”
Figure 37shows (a) the population, (b) the number of retailers, and (c) the number of
“established” retailers who have been operating for five turns or more, in a HowittClower barter economy. There are five goods; five hundred agents, one hundred of
which produce each good; each agent has a productivity of one; each agent can remember
a maximum of fifteen retailers; and the model runs for 200 turns.
155
Figure 37: Equilibrium size of the retail sector
Clearly, the size of the retail sector thus exhibits a rough stability. At any given time,
about fifty agents, or 10% of the population, are engaged in retail trade. This number
fluctuates within a small band but exhibits neither a long-term trend nor a high degree of
volatility. Turnover in retail is limited: at any given time, most of the retailers have been
in business for five turns or more.
Efficiency.Figure 38shows an economy with the same parameters as Figure 37, but tracks
different aggregate variables. Capacity refers to the total productive capacity of the
economy, after taking into account that agents who are engaged in retail can only engage
156
in ordinary production part-time. Consumptionrefers, in this case, to the quantity of
target or consumable goods that agents manage to acquire: if any of the make goods
remain in agents’ hands, they are considered waste.
Figure 38: The near efficiency of a Howitt-Clower economy
For these specifications, the Howitt-Clower “barter” economy is quite efficient, with
consumption generally over 90% of capacity, while some of the remainder is being
invested in retailer inventories. (All goods are perishable when held by agents but can be
157
preserved indefinitely by retailers.) A small fraction of goods goes to waste because
willing sellers and willing buyers fail to connect, or because the retail sector gets the
prices wrong, but most of the goods find buyers and are consumed.
The economy is a “barter” economy in the sense that there is no “legal tender”—retailers
are not bound by law to use an official currency in all transactions—but also in the sense
that a monetary system is not self-organizing, or is self-organizing to only a limited
extent. This is shown in Figure 39, which displays graphically the persistent barter
character of a Howitt-Clower economy. The economy resembles those shown above,
except that it has 50 goods, with 500 agents, with 10 producing each good. Each of the
nodes in Figure 39 represents a good, while each of the edges represents a retailer, the
nodes connected by the edge being the goods which the retailer trades. Figure 39 shows
the trading structure of this economy after 500 turns.
158
Figure 39: The persistent barter character of a Howitt-Clower economy
In Figure 39, as in Figure 36, the circles represent goods, the lines trade offers. Clearly
this is far from being a monetized economy, and the non-emergence of money leads to a
steep fall-off in the efficiency of the barter economy as the number of goods increases, as
shown in the red series inFigure 40. 26
26
Efficiency, in the Howitt-Clower case, means the quantity of goods used over the sum of the quantity of
goods used and the quantity of goods wasted by being “consumed” by an agent who does not like them.
Those wishing to replicate this model should be warned that it is sensitive to the rate at which retailers
159
Figure 40: Barter vs. money
The black series in Figure 40 shows the performance of a money economy, the
implementation of which will be explained in the next section. The superiority of the
money economy explains why every advanced economy is organized not on a barter
basis but on a money basis, and from here on we will focus on monetary economies.
However, our sojourn in the barter economy, a necessary step in adapting the Howittenter. This rate needs to be higher for this experiment than in the endogenous division of labor economy
introduced in the next section.
160
Clower retail sector to be a substitute for the Walrasian auctioneer, has yielded an insight,
namely, the link between money and specialization. In a primitive economy where only a
few goods are produced, a barter economy may be as efficient as a money economy. As
the level of specialization increases, the role of money in eliminating the need for a
“double coincidence of wants” becomes essential.
C. Enter Money
A new good, “money,” is now introduced which has the following special properties: (a)
it is in every agent’s utility function (with a positive coefficient), (b) it is not perishable
(even when agents hold it) or consumable, and (c) one of the two goods that new retailers
trade in is always this good. The interpretation of money being in every agent’s utility
function is not that they enjoy having money per se but that they anticipate future
benefits from holding money, and take these into account in their decisions.
For an agent with a Howitt-Clower utility function, which makes only one good and
values (besides money) just one other good, optimization with money is very simple,
since every path from the production good to the target good runs through money. He
has only to find the highest buy price for his production good, sell it, then find the lowest
sell price for the target good, and buy it. If he encounters stockouts, he will turn to the
next highest-price buyer, or lowest-price seller, continuing until his stocks, or his trading
opportunities, run out.
161
D. How the Howitt-Clower Retail Sector Mimics the Walrasian Equilibrium
In an economy characterized by Howitt-Clower production and utility functions, there
exists a market-clearing price vector. We will call this the Walrasian solution, and how
to derive it is explained in Appendix A. We are now ready to see if the Howitt-Clower
retail sector can reproduce the Walrasian price vector, which in this case is the only price
vector that can clear the market and avoid waste.
Consider an economy with five goods and a larger number of agents 27 randomly assigned
to make and like goods with equal probability so that the aggregate income of all buyers
of each of the goods is roughly the same. Productivity, however, differs among goods:
producers of x 1 receive 1 unit per round, producers of x 2 , 1/2, of x 3 , 1/3, of x 4 , 1/4, and of
x 5 , 1/5. Walrasian equilibrium requires (roughly)that p2 / p1 = 2 , p3 / p1 = 3 , p4 / p1 = 4
, p5 / p1 = 5 . Figure 41shows how prices evolve when these productivity parameters are
used in a simulation with a Howitt-Clower retail sector.
27
In the simulation shown in Figure 8, n=5,000.
162
Figure 41: How the Howitt-Clower model (with money) converges to Walrasian prices
The straight lines inFigure 41 represent Walrasian (normalized relative) prices; the
fluctuating lines represent the time paths of buy and sell prices of different goods (with
the buy price below the sell price); and each color is associated with a particular good.
The buy price is calculated as the total amount of money paid by retailers for the good
divided by the total amount of the good bought by retailers. The sell price is calculated as
the total amount of money spent by agents on the good divided by the total amount of the
good acquired by agents in return. Initially, prices depend on the accidents of early entry
and price experimentation and rise or fall to levels far from their equilibrium values.
163
Over time, however, retailers adapt to supply and demand conditions, and prices adjust
until they closely approximate Walrasian levels.
The Walrasian auctioneer, with no real-world counterpart, must be interpreted as a
personification of the operations of an ideally efficient market, and regarded less as a
theory of how markets work than as a placeholder for a more realistic, though as yet
unknown, theory of markets which would dispense with this unreal entity. HowittClower retailers, though it may seem odd that they trade only two goods and do not
maximize profits, bears a clear enough resemblance to real-world economic actors to be a
plausible interpreted as representing real-world traders. At the same time, the HowittClower retail sector, reassuringly, roughly reproduces the Walrasian result in situations
where a Walrasian solution exists. Meanwhile, this market model can also be applied to
situations where Walrasian equilibrium does not exist.
III. Endogenous Division of Labor: The Agent’s Problem
The Howitt-Clower utility and production functions have served so far as a placeholder
for new functional forms which we will now introduce in order to model Smithian
li − l0i if li ≥ l0i
growth. We give production functions the form mi = 
, where l 0i is a
0 if li ≤ l0i
good-specific fixed cost 28 of production. At the same time, we introduce the utility
28
More strictly, l0 represents an “avoidable cost,” since it is incurred only if the agent produces a positive
amount of the good in question.
164
1/σ


U  M σ + ∑ ai xiσ 
function=
i∈V


, 29 where M is the quantity of money an agent has, a i
measures the agent’s taste for good i, x i is the quantity of good i consumed by the agent,
σ measures the marginal rate of substitution among goods, and V is defined, with a bit of
innocuous vagueness, as the set of all relevant goods. As simple as these functional
forms are, they contain great potential for heterogeneity among goods in the free
parameters a i and l 0i , as well as manipulation of the strength of the taste for variety
through the parameter σ. Also, they create tension between the fixed costs that will tend
to lead agents to specialize in production one or a few goods, and agents’ taste for variety
in consumption. Agents are less dependent on trade than in the Howitt-Clower case, in
the sense that they can achieve positive utility even in autarky, but specialization and
trade promise to increase utility by helping agents reduce overhead and/or diversify
consumption.
Because of fixed costs, the agent’s problem typically does not have an interior solution
but must be treated as an intramarginal problem nested inside an inframarginal problem.
An example will show what these terms mean. 30 A college student must first decide
which classes to take—an inframarginal problem—after which he must decide how much
to study for each class—an intramarginal problem. Similarly, agents (this applies even in
autarky) must first decide which goods to make at all—an inframarginal problem—and
29
While agents include money in the utility function for decision-making purposes, since this represents
saving for the future rather than enjoyment, reported utility will always omit money.
30
I believe I read this example in the work of Xiaokai Yang, but I have not been able to relocate the
citation.
165
after that, how to allocate their labor among them. Since the number of production sets is
finite, agents could in principle explore all of them in succession, but since this is
computationally costly, methods have been developed to expedite the search, which those
willing to wrestle with set theory and algorithm design may study in Appendix C.
The retail sector, which requires no modification when the production and utility
functions are altered but is able to adapt (though one parameter was adjusted 31), changes
the agent’s optimization problem by offering her a set of buying and selling
opportunities. Agents are not “price takers,” both because they face spreads between buy
and sell prices, and because employers and sellers have limited inventories of money and
goods. A graphical and intuitive elucidation is attempted here one crucial and difficult
aspect of the agent’s optimization problem: stockout and—to coin a term—“jobout”
constraints. Stockouts and jobouts are one of the features of the model that corresponds
to the “coordination costs” discussed in Becker and Murphy (1992).
Suppose Safeway offers Brussels sprouts for $2.99/lb. At that price, I would like to buy
five pounds, but they only have two. I could go to Shoppers and buy more, but there they
cost $3.99/lb. I might buy one pound for that price, but after buying two pounds at
Safeway, I no longer value Brussels sprouts enough at the margin to pay that much. This
is a stockout constraint. I earn the highest wage when I sing for weddings. If I could
sing for weddings all the time, I would earn twice as much as I do. But weddings (at my
31
Agents who open retailers are allowed to enjoy 90%, rather than 50%, of their former productivity. This
seemed to make the results more interesting by helping the retail sector to emerge faster.
166
church) only happen occasionally, so I usually work in an office for a lower wage. On
the analogy of the word “stockout,” we may call this a jobout constraint. Stockouts are
inventory constraints on the supply side: a retailer cannot sell as much as I would like to
buy at his price. Jobouts are inventory constraints on the demand side: a retailer cannot
buy as much as I would like to sell him for the price, or wage, he offers.
Figure 42depicts, using a spiderplot, the consumer’s problem for an agent operating in a
multi-good space who faces stockout constraints on his purchases of various goods. The
center of the spiderplot represents the origin, that is, zero consumption of all six goods.
Each of the six axes pointing outwards from the origin shows consumption of a particular
good. The ticks that cut the axes—henceforth, “nodes”—represent the points at which
one seller of the good in question is stocked out, and the agent has to turn to the next
seller. At each of these nodes, a discontinuous price change occurs, as the agent can do
no further business with one seller, and must turn to another seller who charges a higher
price. (In Figure 42, good 6 is self-produced and will never be purchased in the market,
while for good 4, though it is also self-produced, there is one seller whose price is good
enough to make the good worth buying in the market up to a certain margin.)
167
Good 6
Good 5
P5=5.8
Good 1
P1=3.9
P5=5.5
P1=2.7
P5=3.7
P5=3.2 P1=1.1
P4=0.2
P4 is selfproduced
Good 4
P2=77
P3=0.2
P3=0.9
P2=78
P2=79
P3=3.7
P3=7.1
Good 2
P3=16
Good 3
Figure 42: Stockout constraints in consumption-vector space
A consumption bundle may be represented here as a shape that intersects each of the six
axes at points whose distance from the origin corresponds to the quantity of the goods
consumed.
168
Good 6
PRICE
TAKING
ZONE
Good 5
Good 1
SUPPLY BY
INTRAMARGINAL
SELLERS
Optimal
consumption
vector
Good 4
Upper bound
of price
taking zone
Lower bound
of price
taking zone
Good 3
Good 2
Figure 43: A feasible optimal consumption vector under stockout constraints
InFigure 43, three vectors are shown in this six-dimensional goods space: (a) the lower
bound of a “price taking zone,” (b) the upper bound of a “price taking zone,” and (c) an
optimal consumption vector, which is taken to represent the solution to a utility
maximization problem, given the price vector that obtains between these lower and upper
bounds. Specifically, the points in (c) represent the points on the lower bound plus
consumption increments ∆xit , chosen to solve:
(7)

σ
t
t σ
max=
U
M
M
x
x
+
∆
+
+
∆
(
)

(
)
∑
t
t
i
i

∆M t , ∆xt
i∈( Bt ∪ Pt )

169
1
σ
∆M t
Lt
+ ∑ ∆xit Λ ti
 s.t.=
wdt i∈( Bt ∪ Pt )

where M t is the agent’s money stock at the end of the previous step in the iteration, Λ ti is
the “labor price,” 32 direct or indirect, of good i at this margin, wdt is the marginal wage,
and ∆M t is the amount of money the agent plans to add to his money stock (it may be
negative). Crucially, in Figure 43, the optimal consumption vector lies entirely within the
price taking zone, between the lower and upper bounds. That means that when prices are
those that obtain in the price-taking zone, the agent wants to consume a quantity
consistent with those prices. But this will not necessarily be the case. In fact, for any set
of price functions, there will be at most one price taking zone, and there may be none,
which yields an optimal consumption vector that is consistent with the corresponding
capacity constraints. (If there is none, the agent must “hold” some goods at the margin.)
32
While the “labor price” is a convenient conceptual device for understanding the problem of a producerconsumer who is engaged in both market labor and home production—and very much in the spirit of Adam
Smith, who wrote that “the real price of everything, what everything really costs to the man who wants to
acquire it, is the toil and trouble of acquiring it”—it is not used in the exposition in Appendix C, because
the labor prices of buyable goods are undefined in the “underemployment” case where the marginal wage is
zero.
170
W
Job 1
Job 2
Job 3
L
H
Figure 44: With jobout constraints, the wage becomes a function of hours worked
The spiderplot permits some representation of a six-dimensional space, but at a cost:
Figure 43takes the production set as given and does not show prices. Jobout constraints
are in one respect simpler to display, since the agent is interested, not in varied
consumption, but only in money. 33Figure 44represents the employment decision of an
agent who has opportunities to work for several different employers, most of which,
however, are not able to employ him full-time. The vertical axis represents the wage, the
horizontal axis, hours worked. L is the total amount of time available to the agent.
Importantly, the agent cannot devote his full time endowment L to market labor, because
33
Of course, in reality people value jobs for many other things besides money, but that complication is
beyond the scope of this paper.
171
he must incur the overhead costs associated with any job. He may also wish to engage in
some home production as well as market work.
Which of the three work plans shown is best would require far more information than is
in the chart. If the agent wants to engage in a lot of home production, he might prefer job
1, which offers the highest wage for a smaller input of market labor. If the agent wants to
work in the market a bit more, he may opt for job 2, which offers almost as high a wage
at low effort levels and much higher wages when the effort level is somewhat higher.
Yet if the agent wants to do still more market work, he may again prefer job 1. Job 3
seems much inferior, but it cannot be ruled out since the chart does not show overhead,
and it may be that the low wage paid for job 3 is offset by very low overhead. It may also
be that job 3 involves making a good which the agent would wish to make anyway, since
he wants to consume it and its market price is very high. In that case, job 3 would
involve no extra overhead, and might on that ground be preferred to the other jobs, even
though they pay more at the margin.
What can be discerned from Figure 44, however, is that market size matters. The agent’s
decisions are likely to be affected by the fact that he cannot work as much as he likes at
any job without suffering a fall in wages. If the market, and inventories of all retailers in
it, were to grow by a factor of ten, so that jobout constraints ceased to be an issue, the
agent might switch from job 1 to job 2, or from job 2 to job 1. If so, such career switches
would be reflect, and contribute to, Smithian growth.
172
IV. RESULTS I: DIVISION OF LABOR
A vast variety of simulated economies can be generated using the simulation design here
outlined (and described in more detail in the Appendices), and to try to describe all the
possible emergent features of these economies would be a task almost inconceivably
formidable. However, this variety comes from varying only a few parameters, mainly the
number of goods and agents and the variables l 0i ,a i , and σ which define the technology,
and almost any selection of these parameters will illustrate in some way the main thesis
of this paper, concerning the division of labor.
To begin with, consider results derived from a run of the model with ten goods (i.e., nine
goods, plus money) one hundred agents, and σ=0.35, which was run for five hundred
turns, with the technology shown inFigure 45.
173
Figure 45: Technology
The technology shown in Figure 45, which underlies the simulation results to follow, was
randomly generated and is fairly typical. A few guesses can be made just from looking at
the technology about how the economy might evolve. Good 5 is an exceptionally
appealing good, since it is very easy to make (l 05 =0.011) and is well liked by agents
(a 5 =0.667). Another appealing good is Good 1, a bit more difficult to make than Good 5
(l 01 =0.154), but still with fairly low overhead, and is the best liked (a 1 =0.877) of all the
goods. Goods 6 and 8 are almost as well liked as Good 1 (a 6 =0.817, a 8 =0.85), but are
hard to make, suggesting that they are unlikely to be home produced in autarky but may
be strong candidates for commercial production. At the other extreme is Good 7, which
is very easy to make (l 07 =0.003) but contributes little to utility. Good 9, which is both
174
harder to make and less valued than Good 7, seems like it may not be worth producing at
all, and similar doubts may be entertained about Goods 2 and 3, which are a bit better
liked, but harder to make. Good 4, though hard to make and much less popular than
Goods 1, 5, 6, and 8, is moderately liked and might be worth producing on a commercial
basis.
Sales Volume of All Industries, Moving Avg.
30
Industry 2
25
Industry 3
20
Sales
Industry 1
Industry 4
15
Industry 5
10
Industry 6
Industry 7
5
Industry 9
0
0
50
100
150
200
250
300
350
400
450
Industry 9
Turn
Figure 46: The emergence of market-oriented industries
Figure 46, which shows nine time series representing the 11-turn moving averages of
sales by industry, confirms most of these guesses. The first industry to emerge is
Industry 5, and it continues to be one of the two largest industries for most of the
175
simulation, though occasionally eclipsed by Industry 6. Industry 1 takes longer to
emerge, but once established, it soon becomes about as large as Industry 5. It is more
volatile, however, reflecting the fact that Good 5 is constantly being home produced in
any case, whereas Industry 1 has to get wages right in order to persuade workers to make
a significant sacrifice to specialize in it. Industries 6 and 8 also become fairly large,
though their sales remain a bit lower than those of Industries 1 and 5 because of higher
prices. There is no obvious reason why Industry 6 is established so much sooner than
Industry 8. Industry 4 is established early and maintains a consistent presence in the
economy throughout, but is much smaller than the four major industries. Industries 2, 3,
7, and 9 occasionally appear on a small scale but remain small and tend not to survive for
long, though in fact an examination of agent records shows that Good 7 is regularly home
produced.
176
Sales
Sales Volume of All Industries
20
18
16
14
12
10
8
6
4
2
0
Industry 1
Industry 2
Industry 3
Industry 4
Industry 5
Industry 6
Industry 7
Industry 8
400
410
420
430
440
450
460
470
480
490
Industry 9
Turn
Figure 47: Industry sales are quite volatile
The moving average shown in Figure 46suppresses a good deal of volatility, as may be
seen inFigure 47, which shows the raw data for a 100-turn period. Volatility would be
less surprising if Figure 47showed the time series of market production of each good.
The production series, in fact, is even more volatile, but since retailers can store goods,
there is no particular reason production should not be volatile, as odd as that may seem.
(Various factors that make such volatile production inefficient in the real world have
been left out here.) But the utility function gives agents an incentive to smooth
consumption, and since goods are perishable, consumption can be smooth only if sales
are smooth. On the other hand, agents do have the option of home production, which
may help to explain the volatility of sales in some cases. The home production option is
177
especially important for Good 5, with its low fixed costs, the effects of which are
displayed inFigure 48.
Sales and Price in Industry 5
25
Sales
6
Price
5
4
15
3
10
Price
Sales
20
2
5
1
0
0
1
51
101
151
201
251
Turns
301
351
401
451
Figure 48: The home production option makes prices in Industry 5 less volatile, sales more so
Just as the option of easy home production pins makes demand for Good 5 highly elastic,
demand for Good 1 is less elastic, since it is harder to home produce, as shown inFigure
49.
178
Sales and Price in Industry 1
Price
Sales
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
15
10
5
0
150
200
250
300
350
400
450
Turns
Figure 49: Demand for Good 1 is less elastic, making price more volatile, sales a bit less so
While visually, the negative correlation between prices and sales may be easier to see in
Figure 49, in fact it is weaker for Industry 1 (-0.34), than for Industry 5, (-0.63). Greater
dependence of consumers on market production for Good 1 seems to make the industry
more prone to chaotic and cyclical disturbances.
From the industry sales volume time series, it is clear that the economy undergoes a
process of development in the first two hundred turns or so, during which the major
industries emerge one by one, after which, despite continuing fluctuations and
disruptions, a sort of rough steady state becomes established. Figure 50shows another
aspect of this process of development: the displacement of home production by market
labor.
179
Price
Sales
20
Uses of Labor
1
0.9
0.8
Market Work
0.7
Labor
0.6
0.5
Home
Production
0.4
Overhead
0.3
0.2
0.1
0
0
50
100
150
200
250
300
350
400
450
Turns
Figure 50: “From subsistence to exchange,” or, how market labor displaces home production
During the first fifty turns of the simulation, agents devote more than half their labor to
home production. During the last two hundred turns, after all the major industries have
emerged, agents devote less than one-quarter of their labor to home production. Perhaps
surprisingly, overhead labor does not decrease. While this reflects in part the accidents of
the technology, there is a general reason why we should not always expect overhead to
fall with economic development, namely, that while agents mitigate fixed costs by
specializing, they may increase fixed costs by diversifying into high-fixed-costs goods,
rather than settling for “low-hanging fruit,” as they do when the market is less developed.
Figure 51shows what happens to utility as agents shift from home production to market
employment.
180
Utility
6
Avg. U
Median U
Market employment rate
0.6
5
0.5
4
0.4
3
0.3
2
0.2
1
0.1
0
Share of Labor For Market Sales
Utility and Market Employment in an Endogenous Division of Labor
Economy
0
0
100
200
Turns
300
400
Figure 51: How the shift to market employment raises utility
In the first fifty turns, around 15% of agents’ time is devoted to market employment, and
utility is around 1.5. Later, when 30-40% of agents’ time is devoted to market
employment, utility is about 4. That utility continues to be so volatile is counterintuitive, as it is contrary to the traditional belief in “consumption smoothing,” yet it
should be recalled that ordinary people do things like take vacations and celebrate
Christmas which seem designed to concentrate enjoyment at certain times. (Enjoying
oneself, too, may have “fixed costs” and “overhead!”) In the simulation, agents
sometimes alternate between times when they are engaged in market work, and saving,
with low utility, and times when they are engaged in home production, running down
their cash reserves, and enjoying high utility. That said, even with one hundred agents,
181
one might expect that the law of large numbers would do more to erase anomalies in
individual behavior than it does.
If we suppress volatility by taking moving averages of the three series, two tantalizingly
suggestive results emerge, shown inFigure 52. First, the rise in utility lags well behind
the emergence of market specialization—a pattern reminiscent of the common perception
among historians that while the Industrial Revolution has been of great long-term benefit
to mankind, it was initially a mixed blessing. Second, the movements in market
employment track the movements in utility, as seems to happen in real world business
cycles.
182
Avg. U
5
Median U
Market employment rate
4.5
0.4
4
Utility
0.45
0.35
3.5
0.3
3
0.25
2.5
0.2
2
0.15
1.5
0.1
1
Share of Labor For Market Sales
Utility and Market Employment in an Endogenous Division of Labor
Economy, Moving Average
0.05
0.5
0
0
0
100
200
Turns
300
400
Figure 52: Utility tracks market employment closely
However, our priority is not to study business cycles but to implement Smithian growth.
Having escaped the Walrasian trap with the help of Howitt and Clower, having designed
and implemented procedures for agents rationally to adapt to a world with many goods,
fixed costs in producing each good, a taste for variety in consumption, and a HowittClower retail sector, we have now arrived at a model market economy with an
endogenous division of labor. This model economy seems to converge to a steady state,
which allows us to engage in a form of “comparative statics” by adjusting particular
variables and comparing the resulting steady states. However, before testing our main
thesis, it turns out that one more minor modification to the model, though not necessary
183
in principle, is necessary in practice in order to keep run times for the simulation within
reasonable limits. I mention this modification here not only for full disclosure of the
content of the model, but because it has an interesting economic interpretation: learning
by doing.
Figure 53shows the pattern of specialization over time for one randomly chosen agent
(most other agents would look very similar to this one). Initially, the agent steadily
specializes in Goods 1, 5, and 7, but soon he begins to alternate between this autarkic
optimum and a market specialization in Good 6. Later, the agent starts switching to
Good 8 some of the time, with occasional forays into making Good 3, and once each,
tries his hand at Good 4 and Good 9.
How an Individual Agent Switches Specializations Over Time
G
o
o
d
s
9
8
7
6
5
4
3
2
1
0
0
100
200
Turns
300
Figure 53: Flexibility of specialization
184
400
1
2
3
4
5
6
7
8
9
Now the odd thing about the agent in Figure 53is that, while he tends to be specialized at
any given time, over time he switches specializations frequently enough that in the course
of the first 500 turns, he has done every job in the economy, and he hardly ever does the
same job two turns in a row. Of course, this is very unlike the real economy, in which
people tend to have careers, and pursue the same line of work or related lines of work for
an entire lifetime. The reason for this flux is that there is no learning in this economy:
every agent is equally qualified for every job, so small shifts in prices, or jobouts on the
part of particular retailers, or changes in the agent’s wealth, will constantly change which
specialization an agent chooses. For expository reasons, this extreme assumption is
desirable, for two reasons. First, it makes clear that the division of labor is endogenous,
and emerges from the operation of a market and the responses of rational agents to the
incentives it creates. No assumptions whatsoever are necessary about prior distributions
of skills in order for Smithian growth to emerge. Second, it avoids the inevitable
arbitrariness involved in defining learning procedures.
I would continue to leave out learning but for the fact that it is computationally costly for
agents to be able to choose from a very large number of specializations in any given turn.
But it turns out that the complexity of the agent’s decision rises so quickly with the
number of skills (i.e., production options) he has at his disposal that from now on I will
assume that agents can possess at most five skills at any given time. An agent’s skill set
185
at period t depends on his skill sets in previous turns, 34 but also regularly gives agents
new, randomly-chosen skills. This causes agents to adapt a bit more slowly to changes in
the market environment but leaves the main substantive results the same. It makes it
possible to increase the number of goods and to run the economy for more turns. This
done, we can compare several economies which are alike in every respect except the
number of agents.
A new simulation experiment is now introduced, with a new technology, designed
specifically to test the Smithian growth hypothesis. Figure 54shows its main result,
which confirms that “the division of labor is limited by the extent of the market."
34
Specifically, he retains skills he used in the last period with 90% probability; in the 2nd-to-last period
with 70% probability; in the 3rd-to-last period with 50% probability; in the 4th-to-last period with 30%
probability; and in the 5th-to-last period with 10% probability. If that adds up to more than five, one skill,
picked at random, is dropped. It is actually possible, but unusual, for the agent briefly to have more than
five skills. This learning algorithm is partly designed to avoid a somewhat degenerate result, in which the
emergence of markets leads to disaster because it causes people to forget the skills that were most useful in
autarky. I found it uncongenial because, although I did not design agents to take the future into account
explicitly, it would be extraordinarily irrational for agents to forget their old skills immediately as soon as
market work becomes available, and I wanted agents to be roughly rational. That said, the result I avoided
by having agents retain skills longer (at least with a high probability) has an interesting and even plausible
economic interpretation, namely, that the emergence of market exchange can make agents worse off by first
luring them into forgetting survival skills, and then leaving them in the lurch when the instability of a
nascent market causes jobs to disappear.
186
Population and Utility
N=30
N=80
N=200
180
160
Utility
140
120
100
80
60
40
20
0
0
50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
Turn
Figure 54: A larger market leads to higher utility
Figure 54shows how a higher population leads to higher utility in economies with the
same parameters, 35 except that the first has N=30, the second, N=80, and the third,
N=200. In each case, utility is initially very low, because the simulated economy is
“born” without a retail sector to mediate gains from trade, and it takes time for the retail
sector to emerge, and also (now) for agents to stumble upon and put to use the skills that
are most valuable. As retailers emerge and agents find good skill sets, the economy at
first grows quite rapidly, then arrives at a sort of steady state, subject to continuing but
generally small fluctuations. The steady states, however, are at quite different levels,
The parameters include: 50 goods, σ=0.35, maximum five skills per agent at any given time, agent
memory capacity 150 retailers.
35
187
depending on the population. Utility in the N=30 economy hovers around 20. Utility in
the N=80 economy seems to stabilize at above 70. And in the N=200 economy, utility is
sustained at a level of roughly 140. What explains the greater (per capita) prosperity of
the larger economy? It must be the economy’s size, since nothing else was changed, but
is this because “the division of labor is limited by the extent of the market,” or for some
other reason?
188
Median Per Capita Labor in the Steady State, by Industry
N=30
0
0.02
N=80
0.04
0.06
N=200
0.08
GOOD 23
GOOD 35
GOOD 37
GOOD 20
GOOD 22
GOOD 4
GOOD 1
GOOD 8
GOOD 5
GOOD 28
GOOD 14
GOOD 46
GOOD 3
GOOD 45
GOOD 33
GOOD 41
GOOD 11
GOOD 26
GOOD 36
GOOD 17
GOOD 44
GOOD 6
GOOD 34
GOOD 38
GOOD 24
GOOD 48
GOOD 13
GOOD 43
GOOD 29
GOOD 40
GOOD 21
GOOD 9
GOOD 2
GOOD 7
GOOD 10
GOOD 12
GOOD 15
GOOD 16
GOOD 18
GOOD 19
GOOD 25
GOOD 27
GOOD 30
GOOD 31
GOOD 32
GOOD 39
GOOD 42
GOOD 47
GOOD 49
Figure 55: Labor diversifies as the population grows
189
0.1
0.12
0.14
Median Per Capita Consumption in the Steady State, by Industry
N=30
0
0.02
0.04
N=80
0.06
N=200
0.08
0.1
0.12
0.14
GOOD 23
GOOD 35
GOOD 1
GOOD 37
GOOD 4
GOOD 20
GOOD 8
GOOD 22
GOOD 28
GOOD 5
GOOD 3
GOOD 46
GOOD 45
GOOD 33
GOOD 14
GOOD 41
GOOD 11
GOOD 36
GOOD 26
GOOD 44
GOOD 17
GOOD 6
GOOD 21
GOOD 9
GOOD 34
GOOD 16
GOOD 38
GOOD 24
GOOD 48
GOOD 13
GOOD 29
GOOD 43
GOOD 10
GOOD 32
GOOD 40
GOOD 2
GOOD 7
GOOD 12
GOOD 15
GOOD 18
GOOD 19
GOOD 25
GOOD 27
GOOD 30
GOOD 31
GOOD 39
GOOD 42
GOOD 47
GOOD 49
Figure 56: Consumption is diversified but falls by less than labor because fixed costs are reduced
190
Figure 55shows how the division of labor does increase with population size. Each of the
bars shows the median per capita labor applied to the production of each good during the
last five hundred turns of each simulation; goods only appear as bars in Figure 55if they
were produced more than half the time in the economy in question. There were fifteen
goods which the N=30 economy usually produced; twenty-one goods for the N=80
economy; and thirty goods for the N=200 economy. With a couple of exceptions, goods
produced by the smaller economies were also produced by the larger economies, but
fewer resources were devoted to them.
Figure 56, the bars in which represent median per capita consumption during the last 500
turns of the simulation for each population size, shows how greater division of labor
leads to greater diversification of consumption. In some cases the gap between labor and
consumption shrinks in the larger economies, which have been able to pursue
specialization further within certain industries and thus reduce per capita overhead costs.
Mostly, however, gains in the larger economies came from the introduction of new
goods, the commercial production of which was not viable (or at least sustainable) in a
smaller economy. Clearly, even when N=200 there is ample room for further
diversification. About twenty goods are still not being produced or consumed in a typical
simulation round, though they are technologically available in the sense that some agents
know how to make them.
191
Overhead labor (how hard it is to make the good)
Population Size and Exploitation of the Technology
Space
Used by N=30, N=80, N=200
Used by N=30, N=200 Only
Used by N=80, N=200 Only
Used by N=80 Only
Used by N=200 Only
Used by None
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.8
0.6
0.7
Utility coefficient (how much agents like the good)
0.9
1
Figure 57: Small economy goes for "low-hanging fruit," larger economies explore more of the
technology space
Figure 57shows how the endogenous-division-of-labor economy explores the technology
space as the population expands. Technologies in the lower right-hand corner may be
described as “low-hanging fruit.” They have low fixed costs, and make a large
contribution to utility. Agents are smart enough to figure out that these goods are the
most worth making, and these are the first industries to emerge in the economy. Some
goods that are less well-liked but have extremely low fixed costs are also produced in the
192
N=30 economy, to provide a little variety. When the population grows to N=80, several
new goods are introduced. These goods are further from the lower-right-hand corner.
One is less well-liked, the others are a good deal harder to make, than the “low-hanging
fruit” of the N=30 economy. In the N=200 economy, still more technologies are
introduced, generally with still higher fixed costs and/or still lower utility coefficients
than those introduced by the N=80 economy.
There are elements of chance and path dependency in the process by which the
technology space is explored when a maximum skill set size is imposed on agents and its
composition is determined partly by a random process, and it is clear from Figure 57that
the N=200 economy, in particular, did not make all the right technological choices.
When good i has lower fixed costs and a higher utility coefficient than good j, the
economy would always do better to produce good i than good j, yet there are clearly
several goods well within the frontier of technology in fixed cost/utility coefficient space
which the N=200 economy is not using. On the other hand, the technologies the N=200
economy produces are clearly closer, in general, to the lower-right-hand corner than those
which it does not. (That there are two technologies which the N=80 economy produces
but the N=200 economy does not, and one which the N=30 economy and the N=200
economy produce, but which the N=80 economy does not, is probably due to chance.)
The manner in which the economy explores the technology space suggests that there will
be diminishing returns to Smithian growth. The “low-hanging fruit” technologies
193
continue to be used, but do not offer further opportunities for growth in per capita utility,
while the technologies that remain unused as the economy grows are less and less
valuable. It may still be worthwhile for a larger economy to exploit them, due to
economies of scale both in the production process and in the retail sector, and, starting
from zero, the marginal utility of consuming them will initially be high despite their low
utility coefficients. So utility will continue to rise, but not by as much. These
diminishing returns to Smithian growth are indicated in the proportionally smaller gains
in per capita utility from raising the population from N=80 to N=200, compared to the
gains from raising the population from N=30 to N=80. To this we will return.
We have now shown that “the division of labor is limited by the extent of the market,”
but are these markets competitive? Have we been able to generate this result only, as
Becker and Murphy (1992) warned, at the cost of making all industries implausibly
monopolistic? The answer depends on whether we are interested in workers or retailers.
If workers, the answer is no. In the N=30 economy, there are fifteen usually-active
industries. Of these, only two have a median specialist count of one. In the N=80
economy, there are twenty-one active industries, only one of which has a median
specialist count of one. Similarly, of thirty active industries in the N=200 economy, only
two are typically monopolized by one producer. This result should not be surprising.
Only in marginal industries should we expect it to be the case that the available work is
just enough to keep one agent busy. As industries expand, they need more workers,
194
dispelling the problem of monopolistic specialists. Because of the granularity of the
technology—specialization cannot be pursued to arbitrary limits in every direction—most
production will take place in industries that are large enough to demand more than one
worker of the same type. Another factor that may be preventing monopoly is overhead in
the retail sector. It may be worth running a shop selling good i until there is enough
demand for good i to keep more than one worker busy. Finally, agents may avoid
specializing in goods for which the market is small because of jobout problems. Retailers
that try to initiate small industries may find that they cannot offer agents enough work to
persuade them to make them willing to cover the fixed costs of working for them.
If retailers, the answer is yes. The median number of retailers in an industry over long
periods of time, after the steady state has been established, is one in almost all cases.
Early periods of competition tend to give rise to a single winner who dominates retail in
the industry thereafter. The reason for is that retailing in any given industry is a natural
monopoly, with revenues rising with volume while costs (overhead labor) remain the
same. Entry occurs, but entrants initially have lower brand recognition and have
difficulty getting prices right. If they do succeed, they will generally drive out the
incumbent rather than sharing the market. Retailers do not exploit their market power by
raising prices. The retail sector could be interpreted as the scene of many parallel
contestable monopolies, or it could be interpreted, like the Walrasian auctioneer, as a
placeholder for a more sophisticated model.
195
Be that as it may, the model shows at most that monopoly may be a problem in retailing
and market-making, on the one hand, and in a few small industries at the technological
frontier, on the other. By avoiding the symmetry assumption, we have also avoided the
implausible conclusion that capitalist economies should be pervasively monopolistic if
the Smithian thesis is to be applicable. The objection of Becker and Murphy (1992) to
Smithian growth is not compelling.
V. CAPITAL
I suggested above that, as is widely assumed in the literature, Smithian growth might be
subject to diminishing returns, might “peter out.” But if we introduce the possibility that
capital can substitute for labor, this result is dramatically overturned. If it is possible for
agents, by saving in one period, to raise their productivity in the next, the dynamics of
Smithian growth are transformed.
Exponential growth is powerful, and counter-intuitive. “If the number of lilies in a pond
doubles every week, and it takes them 24 weeks to fill the pond, how long does it to take
them to fill half the pond?” is a question some researchers use as a quick gauge of the
intelligence of a survey respondent. Most people answer twelve. The correct answer,
twenty-three, is hard to see because it outrages our sense of proportion. The dull
respectability of the neoclassical growth model tames, as it were, the power of
exponential growth. Endogenous division of labor unleashes it. In the neoclassical
growth model, capital cannot drive long-run growth because it is subject to diminishing
196
returns and depreciation. But if diminishing returns to capital in any given production
process are offset by increasing returns at the level of the whole economy from
endogenous division of labor, then capital may be able, after all, to drive long-run
growth. There will still be some limit to the living standards that can be achieved for
given technological specifications. But because capital accumulation can unleash new
technologies, growth potential can be larger by orders of magnitude than the Solow
model would imply.
When I introduced capital into the Smithian growth model, I was looking for the
emergence of steady, sustainable growth. Sometimes that occurs, but I was amazed
(though in hindsight I should not have been) to find, at least as often, swift transitions
between steady states in which utility changed by several orders of magnitude in response
to seemingly small parameter changes. In other cases, economies plunged into poverty
traps. The exact numbers generated depend on values arbitrarily assigned to several free
parameters and should be taken with a grain of salt, but the more general and qualitative
result—that there is no necessary, a priori upper bound on the pace of growth that can be
sustained by mere capital accumulation—should be taken seriously.
Capital is acquired through foregone consumption, is subject to depreciation, and serves
to augment the labor supply of an individual agent. Specifically, labor supply becomes:
(8)
L = 1 + K 0.5
197
Equation (8) is slightly non-standard in that it implies that production is possible without
any capital. This turned out to be necessary to ensure that low-capital economies, or
individuals, do not sink into such poverty that they cannot cover the fixed costs of
anything, and anyway it seems plausible, since people can produce at least something
without any tools. The exponent on capital ensures that, at the level of any individual
production process, there are diminishing returns to using more and more capital. 36
Capital evolves according to the formula:
(9)

=
K t +1 0.9 K t + s  ∑ xit σ
 i∈( B ∪ P )

σ




Where s is the savings rate, and utility is redefined to be U =
(1 − s )  ∑ xitσ
t
 i∈( B ∪ P )
σ

 . What

(9) means is that capital is depleted by depreciation but replenished by savings, where
savings consist of foregone consumption, as in Solow (1956), Romer (1990) and
elsewhere. Modeling capital as foregone consumption ensures, plausibly and
importantly, that the same intensification of the division of labor that improves the
productivity of labor in generating utility also improves the productivity of labor in
creating capital. Savings rates are exogenous, and there are no capital markets.
36
While the “canonical” exponent on capital is 1/3, this (a) depends on Walrasian assumptions about and
constant returns which are inconsistent with the approach taken here, and (b) does not take returns to
human capital into account. The choice of 1/2 seemed good to me but is essentially arbitrary, and the
results do not depend on this precise parameter choice.
198
I report the results of an experiment with N=400 agents, 150 goods, and σ=0.5. Initially,
though capital is a technological possibility, the savings rate is set to zero, so no capital
forms. After 500 turns, the savings rate is raised to 10%, then after another 500 turns, to
20%. The technology is as shown in Figure 58. While partly random, it is designed to
provide some payoff to capital investment. Many goods, including those that contribute
most to utility, cannot be produced at all without capital, since there are goods with fixed
Fixed cost (how hard it is to make
the good)
costs well above the L=1 unit of labor which a worker without capital has to dispose of.
8
7
6
5
4
3
2
1
0
0
5
10
15
Utility coefficient (how much the agent likes the good)
Figure 58: A technology for which some goods require capital to produce
199
20
When the savings rate is zero, some growth is nonetheless achieved by the refinement of
the division of labor, as shown inFigure 58. By turn 500 of the simulation, this source of
growth seems to be petering out. Growth is continuing, but the rate of growth has fallen
from over 7% per turn in turns 1 to 25 to just over 0.1% per turn in turns 475 to 499.
Utility with Zero Savings Rate
2.5
Utility
2
1.5
1
0.5
0
0
25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475
Turns
Figure 59: Smithian growth without capital
Underlying the growth shown in Figure 59is a continuing diversification of production
and consumption. Figure 60shows the evolution of the most important industries in the
economy (a few industries that appear only briefly are omitted). In the first hundred turns
of the simulation, one industry after another emerges. Each industry’s output tends to fall
after an initial surge as new industries attract part of its work force. Later, some
200
industries continue to emerge, or reemerge, while others give way and disappear. By the
last hundred turns, most industries appear to have flatlined, fluctuating but neither
growing nor shrinking, suggesting an arrival at the kind of stable industrial structure that
characterizes a steady state.
The Emergence of Industries
INDUSTRY 4
INDUSTRY 10
INDUSTRY 21
INDUSTRY 33
INDUSTRY 68
INDUSTRY 111
INDUSTRY 5
INDUSTRY 12
INDUSTRY 25
INDUSTRY 37
INDUSTRY 70
INDUSTRY 121
INDUSTRY 7
INDUSTRY 13
INDUSTRY 27
INDUSTRY 53
INDUSTRY 82
INDUSTRY 8
INDUSTRY 16
INDUSTRY 28
INDUSTRY 56
INDUSTRY 96
INDUSTRY 9
INDUSTRY 17
INDUSTRY 30
INDUSTRY 61
INDUSTRY 106
20
18
16
Industry Sales
14
12
10
8
6
4
2
0
1
51
101
151
201
251
Turns
Figure 60: Industrial diversification
201
301
351
401
451
501
Altogether, there are twenty-six industries which are active more than half the time in the
economy with no capital. Figure 61shows that they are mostly concentrated at the lower
end of the quality spectrum, reflecting the lack of capital to produce higher-quality goods
with higher fixed costs.
Quality and Diversity of Products (Moving Avg., Turns
300-499)
12
Industry Sales
10
8
6
4
2
0
1
11
21
31
41
51
61
71
81
91 101 111 121 131 141
Quality Rank of Good
Figure 61: What the economy without capital produces
When agents begin saving, initially at a rate of 10%, capital rises from zero to just over
0.1 within fifty turns, and then seems to plateau for about 150 turns. The economy would
202
seem at this point to have arrived at a steady state, with utility about 80% above its level
before the rise in the savings rate. In fact, not only are utility and capital holding steady,
but there is little change in industrial structure throughout this period. Nonetheless, after
turn 700, with no exogenous change, the economy begins a long period of steady growth,
as Figure 62shows.
Capital and Utility
3
Savings rate
rises to 10%
100
Utility
Capital
80
2.5
2
60
1.5
40
1
20
0.5
0
Capital
Utility
120
0
0
100
200
300
400
500
600
700
800
900
Turns
Figure 62: A rise in the savings rate leads (eventually) to steady growth
Figure 63shows the evolution of (the moving average of) the growth rate over time,
before and after the rise in the savings rate. Before the rise in the savings rate (after the
initial phase of Smithian growth), the growth rate hovers just above zero. After the rise
203
in the savings rate, although growth takes some time to become established, it then
persists at a steady 1% rate.
Per Turn Utility Growth Rate, Moving Average
0.08
0.07
Savings rate
rises to 10%
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01 1
101
201
301
401
501
601
701
801
901
Figure 63: The savings rate and the growth rate are positively correlated
Economic growth contributes to an increase in capital, which enables the economy to
climb the quality ladder, thus fueling further economic growth. Figure 64shows the
quality profile of goods used in the economy during the last 200 turns of the period in
which the savings rate is 10%.
204
Quality and Diversity of Products (Moving Avg., Turns
800-999)
16
Industry Sales
14
12
10
8
6
4
2
0
1
11
21
31
41
51
61
71
81
91 101 111 121 131 141
Quality Rank of Good
Figure 64: Capital accumulation leads to rising quality of goods
All in all, the effects of raising the savings rate to 10% are rather impressive. Indeed, the
time path of utility is reminiscent of the take-off of modern economic growth after the
Industrial Revolution. Though the rise in the growth rate is both delayed and, when it
arrives, modest, because it is sustained, it has raised utility fifty-fold by five hundred
turns and the rise shows no sign of stopping. Yet it is dwarfed by the growth take-off
occurs when the savings rate rises to 20%, as shown inFigure 65.
205
Capital
80000
70000
60000
Utility
50000
Savings rate
rises to 10%
Savings rate
rises to 20%
40000
30000
20000
10000
0
58
116
174
232
290
348
406
464
522
580
638
696
754
812
870
928
986
1044
1102
1160
1218
1276
1334
1392
1450
0
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
Capital
Utility
Turns
Figure 65: High savings leads to a growth take-off
With a 20% savings rate, growth becomes so fast that within one hundred turns after the
savings rate rise, utility and capital have increased over 400-fold. In order to
accommodate this surge, the vertical axis had to be expanded so much that the growth
that occurred with a 10% savings rate is hardly even visible in the chart. Underlying this
spectacular transformation is a rise in the growth rate, which again follows immediately
after a rise in the savings rate. However, the growth acceleration triggered by the rise to
20%, compared to that which followed the rise to a 10% savings rate, is sharper and more
sustained. Growth surges to 12% per turn, and remains at elevated levels for about one
hundred turns, as shown in Figure 66.
206
Per Turn Utility Growth Rate, Moving Average
0.14
0.12
0.1
Savings rate
rises to 10%
Savings rate
rises to 20%
0.08
0.06
0.04
0.02
0
-0.02 1
101 201 301 401 501 601 701 801 901 1001 1101 1201 1301 1401
Figure 66: Savings rates boost growth
By the end of the simulation, the economy is beginning to exhaust the potential
productivity defined by the technological possibilities of the system. Figure 67 shows that
the economy has already reached the top of the quality ladder and is using almost all of
the most valued goods (while all the goods that were used by the primitive economy
without capital have become obsolete). It can continue to accumulate capital for some
time until, presumably, depreciation and diminishing returns finally put an end to
sustained growth. At that point, something like the Solow steady state will ensue, but
only because of the limits of the technological possibilities in a model that, creating in a
computer, is necessarily finite. In the real world, such limits may not exist.
207
Quality and Diversity of Products (Moving Avg., Turns
1300-1499)
3500
Industry Sales
3000
2500
2000
1500
1000
500
0
1
11 21 31 41 51 61 71 81 91 101 111 121 131 141
Quality Rank of Good
Figure 67: Nowhere else to go on the quality ladder
Surely the most striking result of this experiment is how dramatically economic growth
can respond to changes in the savings rate when, because of fixed costs and endogenous
division of labor, capital accumulation can unleash new technologies. I must admit the
conclusion is, if only suggestively, so optimistic as to be a bit embarrassing. Yet it is not
clear how one would actually make the case that it is implausible. If the living standards
of 2011 could have been foretold to our ancestors two hundred years ago, they would
certainly have seemed incredible. Rates of growth of over 10% have been not only
achieved but sustained by China in recent decades, undergirded by very high savings
rates. Generally speaking, the world economy has not only continued growing decade
after decade, but growth has tended to accelerate over time. The richest and poorest
countries in the world differ by more than two orders of magnitude in their wealth and
208
productivity; the richest and poorest individuals, by much more than that. Smith, who
enthused that the Pin Factory’s division of labor rendered each worker 240 to 4,800 times
as productive at pin making and thought European peasants were far better off than
African kings, would not have found the optimism of these results extravagant. As there
are lots of free parameters, and the technology in particular can be designed to be more or
less friendly to long-run capital-driven growth, there is certainly room for doubt, but the
model should suffice to establish Smithian growth as a strong contender for explaining
both the advance of the economic frontier and the wealth and poverty of nations.
VI. CONCLUSIONS
This paper is, among other things, an attack on the neoclassical growth model of Solow
(1956), in its original form and its textbook incarnations. Solow purported to show that
capital accumulation cannot drive long-run growth, but that result holds only if the
potential gains from specialization and trade have already been exhausted. If there are
gains from specialization and trade still to be had, the facts that (a) capital depreciates,
and (b) capital is subject to diminishing returns, do not in themselves imply any upper
bound on growth. Instead, capital accumulation can drive a growth of economic
complexity that generates aggregate increasing returns, including increasing returns in
producing capital, allowing growth to continue as the space of technological possibilities
is progressively explored. Stigler (1951) said that “Adam Smith’s… famous theorem…
created at least a superficial dilemma.” Yes, but the dilemma is indeed superficial. The
special coordination and agency costs associated with operating in small markets explain
209
why the division of labor will fall somewhat short of the extent of the market, and yet be
limited by the extent of the market, nonetheless. With this theoretical red herring
removed, we are free to raise Smithian growth as an objection to the Solow model’s
starting postulate of a constant-returns-to-scale aggregate production function.
While it is a key finding of the model that capital investment can, after all, drive long-run
growth, this is not done without the help of technological change, in the form of the
introduction of new goods. But here the model stands in sharp rivalry to the theory of
endogenous technological change in Romer (1990). The two theories agree that
economic growth will involve both (a) increased capital, and (b) the introduction of new
goods. But according to Romer (1990), what is needed in order to introduce new goods
is investment in the creation of knowledge or designs. An increase in steady-state capital
will be a mere side-effect of increased productivity due to greater knowledge. That
element might have been incorporated into the Smithian growth story developed here, but
it was deliberately omitted in order to focus on a totally different account of technological
change, namely, that new technologies are introduced when the market is large enough to
make them viable. In short, the model challenges Romer’s interpretation of technological
change.
Strictly speaking, in the model presented above, all technologies are known from the
beginning. There is no discovery whatsoever in the model. Indeed, it will regularly
occur that retailers will open up seeking to trade goods which are far ahead of the
210
economic frontier, only to be disappointed when no one is able to supply them for want
of sufficient capital. The point is not seriously to deny that discovery occurs, but rather,
to suggest that the costs involved in the actual process of invention or discovery may be
trivial, and that what limits the progress of technology is mainly the extent of the market.
It may involve a small investment in R&D to bring cutting-edge technology to market,
but we should not expect the returns to such investment to exhibit the ordinary
proportionality between investment and return, because what really makes the new
technology happen is not the R&D effort of inventing it, but the size of the market being
adequate to render its production profitable. Empirical research can then shed light on
which story is true, or is truer, in general or in particular cases. The Bessemer process,
the Ford Model T (with its capital-intensive assembly-line production process), and the
internet are examples of technologies that seem to have cost very little to invent, but
which could only have emerged in large markets. Prescription drugs are an area where
R&D costs are very high and occur as part of a routine production process, and where, by
contrast, marginal costs of production are low, so that once the brand-name drug has been
invented, it can be copied by cheap generics.
In one respect, the Smithian growth story, though not pessimistic, yet contains a warning.
In the world described by Romer (1990) there is no danger of technological retrogression.
Once the ideas are available, anyone can use them, if the law permits. Ideas apart, the
technology has constant returns. If the world’s population were to shrink by 90%,
economic growth would slow, since fewer people would make fewer new designs, but the
211
fruits of past growth would not be lost, living standards could be sustained. By contrast,
Smithian growth is reversible. If population decline, or a wave of protectionism, or a fall
in the savings rate leading to a depletion of the capital stock, or a deterioration of
institutions worsening coordination and agency problems, were to reduce the effective
extent of the market, some technologies might have to be abandoned, and living
standards would fall.
In other respects, however, the Smithian growth model may be more optimistic than the
Romerian growth model. For example, the Romerian model might interpret China’s
growth performance in recent decades as catch-up growth, which could be sustained at a
high rate only because of continual technology transfer from the West. By this account,
the West, which faces the harder task of pushing the economic frontier forward, cannot
expect to achieve similar rates of growth. By contrast, the Smithian model would
attribute China’s high growth rate to a high savings rate and the opening up of the
country to world trade and investment. The fact that China was borrowing Western
technology is of little importance if the costs of the actual invention process are small
anyway. The West cannot grow as much by opening to trade since it already is fairly
open to trade. But it could open up to immigration, and it could increase its savings rates.
If it did so, suggests the Smithian model, the West might be able to achieve growth rates
similar to China’s.
212
I do not actually wish to maintain that the costs of invention are in all cases trivial, or that
the Smithian growth model is wholly right and the Romerian growth model wholly
wrong. In fact, the two models might fit together nicely. A process of invention, which
in some cases at least is costly and in some cases at least is motivated by profit, supplies
ideas to what has been called the “noosphere,” the sphere of human thought. To put the
ideas to use involves fixed costs, so whether or not this happens, in general and in
particular places, depends on the size of the market. I want to resist, however, the idea
that the bounds of the noosphere, of the space of discovered technology, can be identified
with the economic frontier as represented by the GDP per capita of the most advanced
nations, e.g., the United States. It seems to me quite likely that there are lots of brilliant
ideas out there, yet to be implemented for lack of capital and/or demand, and invisible to
the public, which only becomes aware of technologies when they turn into economic
realities, but well-known to specialists, who might occasionally write their ideas up for
Wired or Popular Mechanics, only to be dismissed as dreamers. I want to leave it an
open question whether the economic frontier is close to the bounds of the noosphere, or
well behind it, with large opportunities for Smithian growth before it pushes up against
the limits of what technologists know how to do.
If Smithian growth is a plausible rival to Romerian growth for interpreting technological
change at the economic frontier, it also holds some promise as a theory of the wealth and
poverty of nations. We must be careful here in how we derive empirical predictions from
the theory. For example, if someone were to suppose that the claim that “the division of
213
labor is limited by the extent of the market” implies that larger countries should be richer,
or should enjoy faster economic growth, than poorer countries, they might object that this
claim is not well-supported by the data. But of course the theory predicts nothing of the
sort, for markets are not co-extensive with countries. More relevant to the theory is
Sachs’ work on geography (e.g., Sachs (1998)), which finds that access to the sea and
transport costs, which determine the degree of access that countries and regions enjoy to
the international division of labor, are a critical determinant of development. This
finding is consistent with Smithian growth, but in general it is a ticklish affair to apply
the concept of “the extent of the market” to comparative development. Infrastructure,
which facilitates trade and communication, and institutions, which facilitate a complex
division of labor by undergirding credible long-term contracts and handling subtle tradeoffs between risk-sharing and incentivization, help to extend the market, but an empirical
study which incorporated infrastructure and institutional quality into an index purporting
to measure “extent of the market” for use as an independent variable in a cross-country
growth regression would, of course, beg major questions about causality. Yet Smithian
growth may serve to inform institutional economics, explaining why institutions matter
and what institutions do more clearly than has hitherto been done, while at the same time
unifying, at least to some extent, economists’ understandings of why all countries were
poor in the past (the long-run growth question) and why some countries are poor today
(the comparative development question).
214
APPENDIX A. WALRASIAN EQUILIBRIUM IN AN ECONOMY WITH HOWITTCLOWER PRODUCTION FUNCTIONS
The goal of a Walrasian auctioneer is to find a price vector that “clears the market,” that
is, that induces equal supply and demand for every good. In the Howitt-Clower setup, the
condition that demand equals supply can be written:
Qid (=
pi )
(10)
∑ y ∀i
j∈Si
j
where Qid ( pi ) is the quantity demanded of good i as a function of p i , its price, S i is the
set of all agents who produce good i, and y j is the productivity of agent j in producing
whatever it produces. Specifically, total demand for good i as a function of prices is:
(11)
=
Qid ( pi )
p (τ j ) 1
=
y
∑
j
pi
pi
j∈Di
∑ p (τ ) y
j∈Di
j
j
whereD i is the set of agents that consume good i, τ j is the type of good that agent j
produces, and p(τ j ) is the price of good τ j . Combining (1) and (2) and solving for p i , we
get:
215
p (τ j ) y j
∑
j∈Di
pi =
=
(12)
∑ y j ∀i
j∈Si
p (τ j ) y j
∑
j∈Di
=
Qis
n
∑p ∑
j =1
yk
(
)
=
Qis
j
k∈ Di ∩ S j
 ∑ yk
 k∈( Di ∩ S j )
pj 
∑
Qis
j =1


n





where Qis , the sum of the individual productivities of makers of good i, is an exogenous
supply parameter for good i.


Now, we may define a set of scalars M ij =  ∑ yk  / Qis . The interpretation of M ij is
 k∈( Di ∩ S j ) 


that it represents the ratio of the quantity of good j that will be brought to market to
purchase good i to the supply of good i. M ij may be zero if no producers of good j buy
 M 11
M
21
good i. Next, we may define a matrix M = 
 ...

 M n1
M 12 ... M 1n 
M 22 ... ... 
. The system of
... ... ... 

... ... M nn 
equations implied by (3) may then be written:
(13)
p = ( Mp ) / ( p ' 1n )
wherep is a vector of prices and 1 n is a vector of ones of dimension n. The last term is
added in order to normalizeprices, since we are interested only in relative prices and the
216
absolute scalar value of prices is a distraction. That aside, the interpretation of (13) is
that equation (12) defines the response of each price to all the other prices, and
equilibrium has been achieved only when the response of each price to all the other prices
is to stay the same. Equation (13) can be converted into an algorithm,
pt +1 = ( Mpt ) / ( pt ' 1n ) , which rapidly converges to the Walrasian equilibrium as t→∞.
217
APPENDIX B. AGENT OPTIMIZATION IN A HOWITT-CLOWER BARTER
ECONOMY
2
1
p=14.4
p=1.1
9
p=0.7
6
p=1.5
10
13
p=10.4
8
p=7.7
11
p=1.2
p=0.2
5
p=1.3
p=0.5
12
p=0.1
7
p=0.7
p=1
p=1.5
4
3
p=1.4
Agent’s Problem
p=19
14
Vertices are goods, edges trade
offers the agent knows about.
Agent has Good 1, and wants
Good 8.
Figure 68: The agent's problem
Figure 68shows an example of the kind of problem that agents in a Howitt-Clower barter
economy have to solve. Each of the vertices (circles) in Figure 68 represents a good in
the economy, while each of the edges (arrows) represents a trade offer of which the agent
218
is aware. An agent A initially possesses some quantity of Good 1. A’s goal is to get as
much as possible of Good 8.
Some of the goods and trade offers, though A knows about them, are irrelevant to his
decision. Thus, the agent is aware of no trade offers involving Good 2, though they may
exist. Similarly, although he is aware of trade offers involving Goods 9, 12, and 13, they
are irrelevant to A, since he can neither acquire any of these goods, nor would they be of
any use to him if he could, since he knows of no way to trade them, even indirectly, for
Good 8. And while A would like to get some of Good 14 if he could, since it can be
traded for Good 8, he has no way to acquire it.
Among the trade offers not already ruled out as irrelevant, the agent is interested in two
particular kinds of paths, which we may call goal paths, that is, paths that lead from
Good 1 to Good 8, and cycles, where one good can be traded indirectly for itself. Figure
69shows the same agent’s problem, but with the cycles highlighted and analyzed.
219
2
1
p=14.4
p=1.1
p=0.7
p=0.2
p=1.5
7
p=0.7
p=1
6
p=0.5
p=0.1
5
p=10.4
8
p=1.3
CYCLE 3
p=7.7
11
p=1.2 CYCLE 2
12
p=1.5
CYCLE 4
CYCLE 1
9
4
3
10
13
p=1.4
p=19
14
Cycles
1. Goods 1, 6. Price: 1.1
2. Goods 9, 12, 13. Price: 0.36
3. Goods 10, 11, 7. Price: 0.91
3. Goods 8, 5, 4. Price: 1.56
Figure 69: Cycles
As Figure 69shows, there are four trading cycles of which the agent is aware, labeled
Cycle 1, Cycle 2, Cycle 3, and Cycle 4. Cycle 1 is a phenomenon that will, in fact, be
encountered very frequently by agents in the Howitt-Clower economy, since retailers
always have two trades on offer, although sometimes one may be inactive because of a
stockout. However, Cycle 1 is of no use to A. What A cares about in a cycle is its
roundabout price, the amount of good i needed to buy, indirectly, one unit of good i. If
this is less than one, the cycle represents an arbitrage opportunity by which the agent can
earn, at no cost, extra goods with which to trade for Good 8. If the roundabout price is
more than one, the cycle is of no use to A.
220
Cycle 2 would represent an arbitrage opportunity, since its roundabout price is 0.36. But
unfortunately, A cannot take advantage of it, because he neither has nor can acquire any
of the goods that are being traded. A is not allowed to borrow goods to which he has no
access by trade. (In this case, the goods acquired by arbitraging Cycle 2 would be of no
use to him anyway, since he cannot trade them for Good 8.)
Cycle 3 also has a roundabout price below one, p=0.91, and A can acquire the goods to
exploit it. Since A’s arbitrage is limited only by the stocks of the retailers along Cycle 3,
these stocks q are shown in Figure 69. The easiest way to see how A maximizes his
arbitrage earnings from Cycle 3 is to suppose that A first identifies the lowest-value
link—in this case, the trade offer that sells up to 2 units Good 10 for Good 7 at a price of
0.5 units of Good 7 per unit of Good 10—and borrows the quantity of Good 10 that is
just sufficient to exhaust this link. Specifically, A will purchase 0.91*2=1.82 units of
Good 10, trade them via Cycle 3 for two units of Good 10, in the process exhausting the
Good 7=>Good 10 trade offer and breaking the cycle, after which he will pay back his
loan, and have 0.18 units of Good 10 left over to be traded for Good 8. Whether A is
permitted to borrow in Good 10 is immaterial, since if not, he can acquire an arbitrarily
small quantity of Good 10 through other trades, and then trade it around the cycle,
increasing his stock, as much as necessary to exhaust the arbitrage opportunity.
Cycle 4 starts from the target, Good 8, and is shown as a reminder that A, if he is rational,
cannot stop searching for paths when he finds a way to acquire the target good, since
221
there may be opportunities to trade the target good itself profitably. In this case,
however, there are not, since the roundabout price of Cycle 4 is greater than one. While
seeking cycles and arbitrage opportunities, the agent is also looking for goal paths, which
are highlighted and analyzed inFigure 70.
2
1
4
3
GOAL PATH 1
p=1.1
9
p=0.7
6
p=2.5
p=1.5
10-
8
p=7.7
11
p=1.2
p=0.2
5
p=1.3
GOAL PATH 2
12
p=0.1
p=10.4
7
p=0.7
p=1
p=14.4
p=1.5
p=1.4
13
p=19
14
Goal Paths
Path 1. Route: 1=>6=>3=>8
Price: 1.1*0.7*14.4≈11.1
Path 2. Route: 1=>10=>11=>8
Price: 0.7*1.4*7.7≈7.5
Figure 70: Acquiring the target good
If the agent did not have separate methods for dealing with cycles, the number of goal
paths would be infinite, since for each of the cycle-free goal paths there would be an
infinite number of other goal paths which looped through one or more cycles various
222
numbers of times. To avoid this, the agent notices when he is going in a cycle, takes note
of it for possible arbitrage later, and meanwhile retreats to the place before the cycle was
completed to look for other paths. Excluding cycles there are in this case only two
trading paths by which Good 1 can be traded for Good 8, identified in Figure 70 as Goal
Path 1 and Goal Path 2. A price is attached to each path by multiplying the prices along
the path. That is, if one trades 1.1 units of Good 1 for one unit of Good 6, 0.7 units of
Good 6 for one unit of Good 3, and 14.4 units of Good 3 for one unit of Good 8, one
ultimately trades 1.1*0.7*14.4≈11.1 units of Good 1 for one unit of Good 8. In this case,
Goal Path 2 is the better option, since by this path one unit of Good 8 costs only about 7.5
units of Good 1.
The agent’s problem can best be conceived in graph-theoretic terms, with goods as the
vertices of a graph, and trade as the edges. The agent’s task is to find all the paths from
the good the agent makes to the good it likes. For an agent that makes good x i and seeks
good x j , the following algorithm solves the agent’s optimization problem:
1.
Create two lists of paths, initially empty, one of “cycles,” one of “goal paths,” a
third list, also initially empty, of edges, called the “stack.”
2.
Identify all the trade offers (edges) for which the pay good demanded (the
starting vertex) is the good the agent makes. Repeat steps 3 through 9 for each of these
edges.
223
3.
Create an active path which is a series of edges, starting with one of the edges
that begin at vertex x 1 . Initially, that is, the active path consists of a single edge.
4.
Look for edges that are potential extensions of the active path, namely, edges for
which the pay good is the same as the sell good for the last edge in the active path. (The
next trade in the sequence must be with someone who wants to buy what the agent now
has, and is offering something for it.)
5.
Add all feasible extension edges to the stack, the purpose of which is to
remember paths that have been identified as feasible but not yet explored. The stack has
the property that one can only add to, or “draw from” the top—that is, when the active
path turns to the stack, the first thing it finds is the last thing it put there.
6.
One of the trade offers from the stack is added to the active path (and removed
from the stack). The agent now seeks to find out whether (a) the active path is a “goal
path” that arrives at a good which the agent wants, or (b) the active path contains a
“cycle,” i.e., turns back on itself.
7.
If the active path is a goal path, the active path is copied into a set of goal paths,
and the exploration continues.
8.
If the active path contains a cycle, the cycle, if it has not been encountered
before, is stored.
9.
In the event of either a cycle or a dead end (not necessarily the same as a goal
path), the agent removes the last link in the path, then checks whether the top trade offer
in the stack is eligible to be placed at the end of the active path (that is, it checks whether
the last sell good in the active path is the same as the pay good for the trade offer at the
224
top of the stack). If so, it adds it and continues. If not, it removes another link in the
active path and sees whether the stack item fits. This continues until the active path has
disappeared.
10.
Check all of the cycles to see if they represent arbitrage opportunities. If so,
exploit them. (How to do this is explained below.) 37
11.
For each good in the agent’s inventory, pick the goal path with the best price and
execute it, converting the good in inventory to the good sought. 38
Arbitrage.In neoclassical economics, arbitrage is often referred to in order to justify the
“law of one price.” The law of one price holds because, if it did not, there would be
opportunities for arbitrage, which would, it is claimed, eliminate the price dispersion.
This argument is used to justify the assumption of perfect competition. In the
neoclassical model, arbitrage seems to play an important role although it never occurs. In
a Howitt-Clower economy, the phenomenon of arbitrage comes to life. (While the
problem of arbitrage becomes quite technical, and while it is necessary, its contribution to
the model is actually limited.)
First of all, two definitions are in order.
37
For computational efficiency, agents are limited to exploiting three loops per turn.)
When there are multiple goods in the agent’s inventory that it wants to trade, it chooses randomly which
of the goal paths to exploit first. This does not matter except in the event of stockouts. In the case of
stockouts, a further optimization problem may arise about how best to use the scarce capacity of
downstream suppliers, but as this would be very complex and rarely relevant or important, it was not
implemented. (In any case, it seems implausible that many customers think strategically about how to use
the limited capacity of the retailers they shop at.)
38
225
DEFINITION 1: A trade cycle is a set of trading opportunities τ 1 , τ 2 , …, τ n , where each
τ i represents an opportunity to buy good x i in exchange for good x (i-1)mod n , at a price
p i , where the units of p i are units of x (i-1)mod n over units of x i .
A trade cycle is an arbitrage opportunity if and only if cyclical trading is profitable,
which depends on prices.
DEFINITION 2: An arbitrage cycle is a trade cycle in which the prices are such that
n
∏p
i =1
i
< 1 , implying that if one unit of a good is traded all the way around the cycle,
more than one unit of the good is acquired.
In order to exploit an arbitrage cycle, it will be assumed that an agent must meet two
conditions:
1.
The agent is aware of the cycle, that is, it knows the identities of the traders
involved, what goods they are ready to trade, and at what prices.
2.
The agent has, or is able to acquire through trade, at least one of the goods which
are traded in the cycle.
As an alternative to condition (2), we might suppose that agents who possess none of the
goods in the arbitrage cycle x 1 , …,x n could borrow some quantity of one of the goods,
226
reacquire the good by cyclical trading, repay the initial loan, and retain some of the
surpluses resulting from arbitrage. We will disallow this on the pretext that credit
markets are imperfect, and in order to contain the complexity of the agent’s optimization
problem.
However, it turns out to be inconvenient and distracting to require that agents who do
have access to the trade cycle actually bring one of the goods in question to market. The
fact is that since transactions are instantaneous and costless, the agent can bring an
arbitrarily small amount to the arbitrage cycle and increase it by an arbitrary amount of
cyclical trading. Instead, agents who have access to the arbitrage cycle do just what has
been disallowed for other agents, namely, they borrow one of the goods, then through
cyclical trading acquire a quantity sufficient to repay the loan and retain a surplus.
For arbitrage to be worthwhile, one or more of the goods which the agent can retain as a
surplus must either be in the agent’s utility function, or be tradable for other goods that
are in the agent’s utility function. There is no need to extend our treatment of the agent’s
problem beyond two special cases: (a) the Howitt-Clower case, in which the agent’s
utility function consists of maximizing a single good, and (b) the “money” case, in which
one special good, “money,” is on one side of every transaction, and an agent’s goal when
engaged in arbitrage is simply to make “money” which can then be allocated optimally to
the purchase of consumption goods. We will focus on the Howitt-Clower case, and how
to deal with the simpler “money” case should also become clear.
227
First, the arbitrageur selects a target goodx j , his surplus of which he will try to
maximize. The choice of x j is trivial in the “money” case (the agent maximizes
“money”), and also in the Howitt-Clower case when the utility good itself is in the
arbitrage cycle. It is also straightforward in the Howitt-Clower case when only one of the
goods in the arbitrage cycle, say x j* , has a trade path to the utility good that lies outside
the arbitrage cycle itself. This is the same thing as to say that no other good has a path to
the utility good except through x j* , and it implies that the agent will seek to maximize the
quantity of x j* in his inventory. If there are multiple paths to the utility good, the choice
of x j is underdetermined, and the arbitrageur will need to explore different options and
see which yields the most utility.
Second, the arbitrageur should identify the bottleneck goodx i* within the arbitrage cycle,
which will be identified by the following condition:
m
(14)
arg min q( j + m ) mod n ∏ p( j +i ) mod n
i* =
( j + k ) mod n, k =
m
i =1
What is distinctive about x i* is that, if the agent begins cyclical trading from good x j , q i* ,
the stock of x i* available for sale, limits from above the quantity which the arbitrageur
can, at the end of the arbitrage cycle, purchase of the target good. This maximum is:
228
b
max =
q bj q=
j*
(15)
qi*
( j −i*) mod n
∏
k =1
p(i*+ k ) mod n
where the expression on the right-hand side represents the quantity of x j which the
arbitrageur will acquire if he obtains all the available x i* and uses it to purchase the
maximum quantities of x (i*+1)mod n , x (i*+2)mod n , …, x j that it will afford him. In order to
acquire q bj * , the agent must borrow the quantity q dj of x j :
( i*− j ) mod n
n
d
b
j
j
i
i*
=i 1 =
k 1
=
q
q=
∏p q
(16)
∏
p( j + k ) mod n
The maximum quantity of the target good which the arbitrageur can acquire through
arbitrage, then, is:
(17)


q = q − q = q 1 − ∏ p  =




1 − ∏ pi 


n
n
b
d
b
i*
j
j
j
j
i
( j −i*) mod n
=i 1 =i 1
( i*+ k ) mod n
k =1
q
π
∏
p
If there are several candidate target goods, the arbitrageur should calculate the surpluses
that could be obtained of each one, then convert each one into the utility good by the
price available along the relevant trade path, and see which choice of x j yields the highest
increment in utility. It is also possible that there are capacity constraints along the path
229
from a proposed x j to the utility good. In that case, the arbitrageur should initially ignore
these capacity constraints when selecting x j , but then should limit qπj to the maximum
that can be converted into the utility good. Having executed this sequence of trades, the
arbitrage cycle will still exist, but the previous x j will no longer be a candidate for the
target good, and the arbitrageur should continue to the next most appealing candidate,
and so on until either the arbitrage opportunity has been exhausted, or there are no paths
left along which goods in the arbitrage cycle can be converted to the utility good.
230
APPENDIX C. THE AGENT’S PROBLEM WITH FIXED COSTS OF PRODUCTION
AND SUPPLY- AND DEMAND-SIDE STOCKOUT CONSTRAINTS
Agents in our Smithian growth simulation face a constrained maximization problem
which in part closely resembles the familiar agent’s problem in neoclassical models, but
in part differs from it sharply. More specifically, the intramarginal aspect of the problem
is thoroughly neoclassical, but there are also inframarginal aspects which neoclassical
economics characteristically avoids by assuming away nonconvexities, but which in the
context of a Howitt-Clower retail sector and a set of fixed-cost production functions an
agent must solve if it is to be “rational,” in the sense of maximizing utility.
The agent has only an instantaneous utility function, not an intertemporal utility function.
The reason for this is that the environment is so complex that it is not feasible to create an
accurate forecast of the future and build it into the agent’s problem. However, the agent
takes the future into account by including money in the utility function, the interpretation
of this being that while holding money is not pleasurable in itself, the agent enjoys the
security of having funds saved up for the future.
The Intramarginal Problem
231
We will deal with the intramarginal problem first, and to do so we must assume that the
inframarginal problem has already been solved. That is, a production set P has already
been chosen. There is also a set B, the “master” or “market” buyable set, which is
exogenous and independent of the agent’s decision, and which includes all the goods that
are for sale in the market. Some goods may be in both B and P, others will typically be in
only one of these sets.
Given P, there may be a good s ∈ P which has the highest sell price, or s = arg max wi ,
i
where w i is the sell price of, or the “wage” the agent earns (at the margin) by producing,
good i. If this good exists, we will call the wage that can be earned by producing and
selling it the marginal wage w t , since it may vary with the stage of an iteration process,
indicated by t. Such a good may not exist if there are no buyers (or none of which the
agent is aware) of any of the goods in P. This can be called the unemployment case, or
better (for greater generality, in case the marginal wage only falls to zero after some
margin) the underemployment case. Unemployed agents still engage full time, however,
in home production, so unemployment refers to a lack of market employment. Related to
this, but distinct, is the autarky case, in which the agent chooses neither to buy nor to sell
any goods. An agent can be unemployed but not in autarky because he has cash reserves
with which to buy goods that are for sale in the market. Conversely, an agent can be
employed (in market production) but self-supply all his consumables, saving his earnings
for the future. An agent might be unemployed even though market jobs are available
because the wages are too low, so that he prefers home production. As the full
232
employment and underemployment cases have somewhat different solutions, we will deal
with them in sequence.
Full employment.In the full employment case, w t >0. Also, for the sake of generality, we
will assume that the agent already possesses a certain quantity of money M, and a certain
quantity xit of each good i ∈ ( P ∪ B ) . At the beginning of a trading process, of course,
xit = 0∀i ∈ ( P ∪ B ) . The quantity of labor which the agent has left after “overhead” labor
∑l
i∈P
0i
and any labor spent in previous rounds of the trading process we will call L t .
Although at this stage P, the set of goods that the agent has chosen to incur the overhead
costs for, is taken as given, it may not be true at every margin, or even at any margin, that
P is also the set of goods which the agent should produce. If i ∈ P but the agent’s best
intramarginal decision is not to produce any of good i, P was incorrectly chosen, but the
intramarginal decision should nonetheless be able to make the best of this bad
inframarginal choice. On the other hand, to include i in P may have been the right
decision even though the agent is better off buying, rather than self-producing, good i at
some margins. For example, if good iis available for purchase at a very low price, but
only in a very small amount, an agent might want to incur overhead costs so as to be able
to self-produce i, but still take advantage of the bargain. We may therefore define
Pt :=
{i ∈ P | i ∈ B, wt ≤ pit } ∪ {i ∈ P | i ∉ B} and Bt := {i ∈ B | i ∉ P} ∪ {i ∈ B | pit < wt } , or in
words, P t (the marginal production set) includes all elements of P (the production set)
233
which either cannot be bought or which are less expensive to make (in the opportunity
cost of time) than to buy, while B t (the marginal buyable set) includes all the elements of
B and not in P, as well as all the elements of B for which the market price is less than the
marginal wage.
Having defined the marginal production and buyable sets, the agent’s maximization
problem, then, is:
1/σ
(18)


σ
t
t σ
t
t σ
=
+
∆
+
+
∆
+
+
∆
U
M
M
a
x
m
a
x
b
max
(
)
(
)
(
)


∑
∑
t
i
i
i
i
i
i
yt , ∆M t , ∆mit ∀i∈P , ∆bit ∀i∈B
i∈Pt
i∈Bt


t
t
t
∆M t + ∑ pi ∆bi , Lt =+
yt ∑ ∆mi
s.t. yt wt =
i∈Bt
i∈Pt
Where the choice variables are y t or market-oriented labor, ∆M t or the change in the
agent’s money holdings, ∆mit or the quantity of good i self-produced (“made”) for all
i ∈ P , and ∆bit or the quantity of good ibought for all i ∈ B , and the parameters are M,
the agent’s money holdings prior to solving the intramarginal problem, w t , the marginal
wage, prices pit , prior holdings of goods xit , and technology-related parameters a i and σ.
Solving (18) first for the optimal value of y t , as a function of the parameters, we get:
234

M t + ∑ pit xit

1
i∈Bt
(19) yt * = Lt + ∑ xit − wt σ −1
σ
1

i∈Pt
t σ −1 1−σ
1 + ∑ pi ai

i∈Bt

1
∑ ai1−σ
i∈P

σ

σ −1
wt
 1 +
σ
1

t σ −1 1−σ
  1 + ∑ pi ai
i∈Bt

1
∑ ai1−σ
i∈Pt






−1
From there, assuming that y t * is feasible, that is that 0 ≤ yt * ≤ ytmax , and if we define a
value κ =
M t + wt yt * + ∑ pit xit
i∈Bt
σ
t σ −1
1 + ∑ pi
i∈Bt
ai
1
1−σ
, the optimal values of the other choice variables may be
stated as follows:
∆M t =κ − M t
1
(20)
 p t σ −1
=
∆bit κ  i  − xit
 ai 
1
 w σ −1
=
∆m κ  t  − xit
 ai 
t
i
Now, the value of κ is always positive, but if some of the xit are also positive, some of
the values of ∆bit and ∆mit might be negative. Also, some of the values of ∆bit might
violate the stockout constraints, and y t might violate the jobout constraint of the marginal
job, that is, y t might involve the agent trying to earn more income from the marginal
employer (or seller) than that employer is able to pay.
235
But if we focus on the special case in which the agent is a “price taker,” that is, in which
no stockout or jobout constraints are binding and the agent can buy or work as much as
he likes without affecting prices and wages, we can ignore these problems for the
moment. Price-taking also implies that the xit are zero, since goods are perishable and
the only reason the agent might have them in inventory is if he is at a midpoint of a
trading process by stockouts or jobouts, and therefore that the values of ∆bit and ∆mit are
positive. 39 In that case, U* can be calculated as:
(21)
σ
σ
1
1




t σ −1 1−σ
σ −1
1−σ
U * ( P ) = M t + wt  L − ∑ l0i   1 + ∑ pi ai + wt ∑ ai 
i∈P
i∈P

   i∈B


1−σ
σ
Where L t has been rewritten as L − ∑ l0i , since we have assumed that jobout constraints
i∈P
are not binding and the intramarginal problem is solved in one round, so that L t equals
the total labor supply minus overhead.
Underemployment. For a fully employed individual, “time is money”: the market wage
fixes the ratio of the marginal value to the agent of money and of labor. For an
underemployed individual, this is not the case. She has “zero marginal product,” at least
as far as the market is concerned, that is, she either had no opportunities to earn money or
39
Strictly speaking, a lack of binding stockout or jobout constraints does not exactly make the agent a price
taker, because the buy and sell prices of goods still differ. On each side of the market the agent is a price
taker, but not overall. He affects the price by his decision to buy, or to sell.
236
has exhausted them, and at the margin, no one is willing to pay her to do anything. At the
margin, time is not (convertible into) money. 40
Consequently, the decision for the underemployed person is in some ways more
complicated than that faced by the fully employed individual. For the fully employed
individual, the question of what to buy and what to self-produce, given that the decisions
to initiate production have already been taken, is straightforward: allocate to P t (make)
that is in the production set and whose market price is higher than the wage; and allocate
to B t (buy) anything else available. An underemployed person cannot use the wage to
allocate goods to P t and B t , yet the idea of equalizing the ratio of the marginal values of
time and money is still the key to making the right decision about what to self-produce
and what to buy. This involves finding the relative shadow value of time and money.
Two cases must be distinguished: (a) the case in which there is a dual-sourcegood, which
the agent both buys and self-produces, and (b) the case in which all goods fall cleanly
into the purchased and self-produced categories. For reasons that will become clear later,
we will call case (a) the “black key” case and case (b) the “white key” case. Consider
first the white key case, which is the simpler of the two cases. Suppose the allocation of
goods to B t and P t has already been made, and there are no dual source goods. The agent
must then solve:
40
This holds even if market work is available which is not worth the agent’s time to pursue, because either
the agent is better off doing without market purchases (and new savings) altogether, or else she prefers to
finance market purchases by running down her savings. In that case, while time can be converted to money
at the margin, the wage at which this conversion would take place is not relevant to the agent, since her
time is worth more to her than the wages that the market is willing to pay her for it.
237
1/σ
(22)

σ
σ 
σ
U  ( M + ∆M t ) + ∑ ai ( xit + ∆mit ) + ∑ ai ( xit + ∆bit ) 
max=
i∈Pt
i∈Bt


t
t
t
s.t. ∆M t + ∑ pi ∆bi = 0, Lt = ∑ ∆mi
i∈Bt
i∈Pt
The solutions to (22) are:
=
(a) ∆M t
M t + ∑ pit xit
i∈Bt
1+ ∑ p
i∈Bt
(23)
=
(b) ∆bit
− Mt
σ
1
t σ −1 1−σ
i
i
a
M t + ∑ pit xit
i∈Bt
σ
1
1 + ∑ pit σ −1 ai 1−σ
1
 pit σ −1
t
  − xi ∀i ∈ Bt
a
 i 
i∈Bt
=
(c) ∆mit
Lt + ∑ xit
i∈Pt
1
1−σ
i
∑a
i∈Pt
1
ai 1−σ − xit ∀i ∈ Pt
In effect, in the white key case, the agent solves two separate optimization problems, one
to maximize the utility derived from the agent’s labor, a separate one to maximize the
utility derived from the agent’s money. Equation (23c) depends only on labor and on
quantities of and tastes for goods in set P. Equations (23a) and (23b) depend only on
money and on quantities, prices, and tastes for goods in set B. While this optimization is
straightforward, it assumes an allocation of goods to sets P and B and provides no
guidance for how to choose that allocation.
238
In the black key case, a good d, such that d ∈ B and d ∈ P , is first chosen, and is used to
define the marginal buyable and production sets Bt := {i ∈ B | i ∉ P} ∪ {i ∈ B | pit < pdt } and
Pt :=
{i ∈ P | i ∉ B} ∪ {i ∈ P | i ∈ B, pit > pdt } . This done, the agent’s maximization problem
is:
max U =
1/σ
(24)


σ
t
t σ
t
t σ
t
t
t σ
 ( M + ∆M t ) + ∑ ai ( xi + ∆mi ) + ∑ ai ( xi + ∆bi ) + ad ( xd + ∆bd + ∆md ) 
i∈Pt
i∈Bt


t
t
t
t
t
t
s.t. ∆M t + ∑ pi ∆bi + pd ∆b=
0, L=
∑ ∆mi + ∆md
d
t
i∈Bt
i∈Pt
Where, as usual, ∆bit is the quantity of good i that each agent plans to purchase and ∆mit
is the quantity of good i that the agent plans to self-produce, but now ∆bdt is the quantity
of the dual-source good the agent plans to purchase, and ∆mdt is the quantity of the dualsource good the agent plans to self-produce. 41 The problem in (24) can be solved for
∆bdt , which is:
41
There will never be more than one dual-source good because if an agent were to both make and buy a
good, she could make herself strictly better off by making more of the more expensive one and buying
more of the cheaper one, until one of the zero bound constraints becomes binding.
239
σ
1
 
t
t 
t σ −1 1−σ 
L
x
x
1
p
a
−
+
+
+
  t
∑
∑ i i 
d
i 
i∈P
  i∈B
 

σ
1
1


+ pdt σ −1  ad 1−σ + ∑ ai 1−σ 
i∈P


1
1
1

t t  t σ −1 
1−σ
1−σ
M
p
x
p
a
a
+
+
∑
i
 t ∑ i i d  d
i∈B
i∈P



t
(25) ∆bd =
σ
1
1 + ∑ pit σ −1 ai 1−σ
i∈B
From (25), optimal values for all the other choice variables can be derived, including
∆mdt . These values are subject to the usual constraints, but the zero-bound constraints on
making and buying the dual-source good, ∆bdt ≥ 0 and ∆mdt ≥ 0 , are especially important.
Crucially, in the black key case, the optimization ensures that the shadow wage must be
equal to the market price of the dual-source good, or in other words, the money needed to
buy a bit more of the dual source good, and the labor needed to make it, must be valued
the same by the agent. The values of ∆bdt and ∆mdt will adjust to put this shadow price
where it needs to be. In the process one or the other of them will typically overstep the
zero bound. If, at the margin, the agent is planning to self-produce too many goods, the
shadow wage will be pinned at a relatively low level by the choice of a dual-source good,
and the agent will wish to “make” a negative quantity of the dual-source good, buying it
in the market to offset the negative production, in order to allocate extra labor to other
production activities. If, on the other hand, the agent is planning to buy too many goods
in the market, the shadow wage will be pinned at a high level, and the agent will want to
240
buy a negative quantity of the dual-source good, offsetting these negative purchases with
positive production, in order to get money to buy other goods. There will be at most one
allocation of goods to these three categories, self-produced, bought, and dual-source,
which yields a decision that satisfies ∆bdt ≥ 0 and ∆mdt ≥ 0 . If such an allocation exists, it
is optimal for that trading step. If it does not, the optimal allocation will involve no dualsource good.
The search for the optimal allocation of goods to the buy, self-produce, and dual-source
categories may be compared to a person trying to find the key on a piano keyboard which
matches a pitch. The pitch corresponds to a shadow wage. Once a shadow wage is
decided, everything follows from it: goods are allocated to the buy, self-produce, and
dual-source categories, and the values of ∆M t and of ∆bit and ∆mit for all goods i are
chosen. A shadow wage equal to the marginal price of one of the goods implies that
there is a dual-source good: we may compare these cases to the “black keys” of the piano.
A shadow wage that does not equal the marginal price of any of the goods implies that all
goods fall cleanly into the buy or self-produce categories: these are the “white keys” of
the piano.
A musician can tell whether the pitch he plays on a piano is higher or lower than the pitch
he is trying to match. The odd thing about the shopping problem in the
underemployment case is that only when a dual-source good is assumed does the solution
provide information about whether the selected shadow wage is too high or too low
241
relative to the optimum—as if a musician could only tell whether he was above or below
the proper pitch when he plays black keys. Accordingly, the search algorithm must focus
on the black keys, keeping track of the highest shadow wage that proved to be too low
and the lowest shadow wage that proved to be too high, and exploring the shadow wages
in between, until either (a) a choice of shadow wage is confirmed by the subsequent
optimization, or (b) two adjacent “black keys” prove to be too high and too low,
respectively, indicating that the “white key” between them corresponds to the right
shadow wage.
The Inframarginal Problem in the Price-Taking Case
The optimal utility U*, in (21), is written as a function of P, the production set.
Implicitly, the wage w t and the buyable set B are also functions of P. For purposes of the
intramarginal problem, P, and therefore w t and B, are taken as given. The inframarginal
problem does not take these as given, but seeks to explore the space of possible
production sets to find the optimal P*.
Since P consists of a set of goods, or more precisely of a Boolean vector with
dimensionality equal to the number of goods which it is possible for the agent to produce,
defining whether each good is in or out of the set, there is a finite total number of
possible production sets P, and it is therefore possible, in principle, for the agent to find
the optimum simply by trying each one in succession. However, as the number of
242
possible production sets rises exponentially with the number of goods, the computational
costs of such a procedure are exorbitant whenever there are more than a few goods
available for the agent to produce.
To find the solution more quickly, we can begin by defining:
(26)
σ
1
t σ −1 1−σ
i
i
A( P) =
1+ ∑ p a
i∈B
+ wt
σ
σ −1
∑a
i∈P
1
1−σ
i
And
(27)


B ( P) =
M t + wt  L − ∑ l0i 
i∈P


After which equation (21) can be rewritten as:
(28)
U * ( P ) = B ( P ) ( A ( P ))
1−σ
σ
Since U* is an increasing function of both A(P) and B(P), we know that for any two sets
P 1 and P 2 , if A ( P1 ) > A ( P2 ) and B ( P1 ) > B ( P2 ) , then U * ( P1 ) > U * ( P2 ) . Now
suppose there is a set P 0 , a good i and a good j such that P=
P0 ∪ i and P=
P0 ∪ j . In
1
2
that case:
243
(29)




B ( P1 ) − B ( P2 ) = M t + wt  L − ∑ l0 k  − M t − wt  L − ∑ l0 k 
k∈P1
k∈P2








= wt  ∑ l0 k − ∑ l0 k=
 wt ( l0i − l0 j )
 wt  ∑ l0 k + l0i − ∑ l0 k − l0 j=
k∈P1
k∈P0
 k∈P2

 k∈P0

That is, whether B ( P1 ) > B ( P2 ) can be discovered simply by comparing l 0i and l 0j . A(P)
is a bit more complicated, because it involves both the sets B and the set P. Now, B is a
function of P, such that, if a good iis available for sale in the market, then the exclusion
of i from P will immediately imply its inclusion in B. If the agent does not make good i,
he will buy it, at least in small quantities, and likewise with good j. We may therefore
define a set B 0 corresponding to P 0 , such that if goods i and j are both available in the
market, B0 = B1 ∪ j = B2 ∪ i . In that case:
A ( P1 ) − A=
( P2 )
∑
=
k∈B0
(30)
∑
k∈B1
σ
σ
1
σ
1
k∈P1
k∈B1
1
1
1
σ
σ
σ
 1

pkt σ −1 ak 1−σ − pit σ −1 ai 1−σ + wt σ −1  ai 1−σ + ∑ ak 1−σ 
k∈P0


1
1
1
σ
σ
σ
 1

− ∑ pkt σ −1 ak 1−σ + p tj σ −1 a j 1−σ − wt σ −1  a j 1−σ + ∑ ak 1−σ 
k∈B0
k∈P0


σ
1
σ
1
σ
σ
1
1
= p tj σ −1 a j 1−σ + wt σ −1 ai 1−σ − pit σ −1 ai 1−σ − wt σ −1 a j 1−σ
= ai
1
1−σ
1
σ
1
pkt σ −1 ak 1−σ + wt σ −1 ∑ ak 1−σ − ∑ pkt σ −1 ak 1−σ − wt σ −1 ∑ ak 1−σ
σ
σ
1
 σσ−1
 σσ−1
t σ −1 
t σ −1 
1−σ
 wt − pi  − a j  wt − p j 




244
k∈P1
Thus, when goods i and j are both available in the market, A ( P1 ) > A ( P2 ) if and only if
1
σ
1
 σ

ai 1−σ  wt σ −1 − pit σ −1  > a j 1−σ


σ
 σσ−1
t σ −1 
w
p
−
 t
 . When good i and/or good j is not available
j


σ
σ
t σ −1
i
t σ −1
j
for purchase in the market, the term p
and/or p
can be omitted, and the
expression can take three other forms. To abstract from these complications we can
define:
(31)
 1
a 1−σ
 i
hi ( wt ) = 
 1
 1−σ
ai
σ

t σ −1 


p
1 − i

 w  
  t 


 the marginal wage wt > 0; and
if 
good i is for sale in the market
otherwise
Where h i is written as a function of w t as a reminder that even in the price-taking case w t
is not a mere parameter but a function of the choice of P. Based on (29), (30), and (31)
we can define a relation
(32)
hi ( wt ) ≥ h j ( wt )

i  s j ⇔ l0i ≤ l0 j
p

hi ( wt ) > h j ( wt ) or l0i < l0 j
The relation in (20) can be written in words as “iis preferred to produce relative to j,
given sell good s.” Importantly, while whether i  s j holds or not depends on the sell
p
245
good (and the wage), it does not depend on any of the other goods in the production set.
The significance of the relation is analogous to that of preferences in utility theory. If
i  s j , that implies that the optimal production set (given sell good s), may include both i
p
and j, or neither i nor j, or i and not j, but it may not include j and exclude i.
The relation i  s j allows us to define, within any set of goods, and always assuming a
p
sell good and a wage, certain subsets of special interest. We can define the “innermost”
subset of goods within a set, I ( P=
):
{i | ¬∃j | j  i} , or in words, I(P) is the set of goods
s
p
in P relative to which no other goods in P are preferred to produce, given the wage;
briefly, the most preferred goods in P. Similarly, we can define the “outermost” subset
of goods within a set, O ( P=
):
{i | ¬∃j | i  j} , or in words, O(P) is the set of goods in P
s
p
which are not preferred to produce relative to any other goods in P; briefly, the least
preferred goods in P. Finally, we can define the “core” of P, C(P), consisting of all the
P and C ( P ) ∩ O ( P ) =
∅.
goods in Pexcept the outermost, that is, C ( P ) ∪ O ( P ) =
Using these three set functions, we can define the following procedure for searching for
the optimal production set P* ⊆ V | U ( P *) ≥ U ( P ') ∀P ' ⊆ V , 42 where V is the exogenous
set of production possibilities available to an agent. First, define a set Q(P) such that
42
Strictly speaking, it is possible that there might be multiple production sets that satisfy the condition, and
P*should be regarded as a set of production sets rather than a single production set, with the agent
searching for any element of P*, it does not matter which. However, as the cases where P* has more than
one element are necessarily rare and exceptional, we will ignore this subtlety.
246
Q ( P) ∪ P =
V ,Q ( P) ∩ P =
∅ ; in words, this is the “inverse” of P within V. Second,
define I(P), O(Q), and X=
( P ) O ( P ) ∪ I ( Q ( P ) ) , which may be called the set of
“candidates” for inclusion in a set superior to P. From the derivation of i  s j , we know
p
that (a) if the agent will not benefit from omitting from the production set any of the
elements of O(P), she will also not benefit from omitting from the production set any of
the other elements of P, and (b) if the agent will not benefit by including any of the
elements of I(Q), she will also not benefit from including any of the other elements of Q.
Therefore, if there is any P ' | U * ( P ') > U * ( P ) , then there is some
Y ⊆ X | U * ( C ( P ) ∪ Y ) > U * ( P ) , and conversely, if such a set Y does not exist, then
U * ( P ) ≥ U * ( P ') ∀P ' ⊂ V , or in other words, P=P*. The agent can find the optimal P*,
then, using the following iteration:
Iteration 1
1.
Select any Pt =0 ⊆ V as a starting point.
2.
Define Q(P t ), O(P t ), C(P t ),I(Q(P t )), and X(P t ).
3.
Calculate U * ( C ( Pt ) ∪ Y ) for all Y ⊆ X ( Pt ) or until some
=
Y ' Y | U * ( C ( Pt ) ∪ Y ) > U * ( Pt ) is discovered.
4.
Pt +1 C ( Pt ) ∪ Y ' , return to step 2, and
If such a Y’ has been found, define=
continue iteration.
247
5.
If no Y’ was found in step 3, P t =P*. End iteration.
Iteration 1 is, in fact, a special case, where no stockout or jobout constraints are binding.
We will see below how it needs to be modified to accommodate the possibility of
stockout and jobout constraints. In any case, Iteration 1 is not a complete solution to the
inframarginal problem because it takes the sell good, and the wage, as given. Instead,
Iteration 1 must be run several times, to accommodate different possible sell goods. The
number of sell goods is finite, and in this case it is less urgent to develop an algorithm to
accelerate the search, since the number of candidate sell goods is n at most. We can
eliminate any good i | ∃j | w j > pi , l0 j < l0i , for in that case it will always be better to selfproduce good j and trade for good i than to self-produce good i. For any goods s not thus
eliminated, Iteration 1A is used to find the optimal P * ( wt ) ⊆ V ' ( s ) , where
V ' ( s ) :={i | i ∈ V , wi ≤ ws } , since any good i such that wi > ws would, if produced,
displace s as the sell good. The procedure for selecting the sell set, however, will also
need to be redefined once the possibility of jobout constraints is taken into account.
Stockouts and Jobouts
If we take stockout constraints into account, all buy and sell prices become functions of
the quantity that the agent plans to buy or sell. It is no longer possible to define a general
labor price of acquiring each good, because all w i and p i are functions of the quantity
(respectively) sold or bought. Instead, the one-off intramarginal problem above is
248
converted into an iterative problem, in which an agent maximizes utility as if she were a
price taker, then checks whether her income and shopping plans are feasible, and adjusts.
For a given production set P, there is a series of nb buyers of goods in P—or, employers
of workers making goods in P—a series of wages wi ,..., wnb , and a series of quantities
z1 ,..., znb , which mark the points where successive employers’ bank accounts are
exhausted, and thus the cut-offs between one wage and the next. These wages are
arranged so that w1 > w2 > ... > wnb , that is, in descending order by wage, since the agent
will work at the highest-paying jobs first. The agent might switch “jobs”—switch her
production good—several times as she exhausts the bank accounts of successive
employers.
s
For each good i which the agent can buy, there is a series of prices pi1 ,..., pini , where nis is
s
the number of sellers of good i, and a series of quantities qi1 ,..., qini , which, again, mark
the points where success sellers’ inventories are exhausted and thus the cut-offs between
one price and the next. In this case, though, the prices are arranged so that
s
pi1 < pi2 < ... < pini , that is, in ascending order by price, since the agent will buy from the
lowest-price sellers first.
Since we know how to solve for U* given a specific price vector, and since wages and
prices are constant over particular ranges, the agent will still ultimately be maximizing a
249
utility function by choosing the quantities to make, buy, sell, and hold subject to a labor
constraint and a labor price vector at the margin. But which margin is relevant depends,
in turn, on the quantities the agent buys and sells. There will be at most one choice of
quantities which is optimal at the corresponding margins.
Suppose, then, that we are continuing an iteration, and at iteration t, the agent has
provisionally decided to produce a set of goods P, to buy, at the margin, a set of goods B t ,
and to produce at the margin a set of goods P t , such that Pt ⊆ P, Bt ∩ P =
∅ . Also, the
agent has decided to exhaust the demand of dt − 1 buyers and of dit − 1 sellers of each
good i, so that at the margin she is selling to buyer (or, working for employer) d t and
{B ; P ; d ; d ∀i ∈ B }
buying each good i from seller dit . We can call this a decision
=
Dt :
t
t
t
t
i
t
, consisting of the buy and sell sets, the marginal buyer, and the marginal seller of each
good i in B t .
The agent inherits from the last iteration a quantity of money
dt −1
dit −1
dit −1
M t =+
M 0 ∑ wk yk − ∑ ∑ p q a quantity of each good x = ∑ qik , and a quantity of
=
k 1
=
i∈V k 1
k
i
k
i
t
i
dt −1
k =1
L − ∑ l0i − ∑ yk . The marginal price of each good is pidi , and the marginal
labor Lt =
i∈P
t
k=
1
wage, wdt , which may be (a) positive, or (b) zero. With this information, the agent is in a
position to set up and solve the full employment optimization problem described in
equations (18)-(20), yielding values for the choice variables, y t , ∆M t , ∆bit , and ∆mit . (If
250
the agent begins solving the full-employment problem and finds that y t *<0, she drops it
and solves the underemployment problem instead.)
Of the choice variables, ΔM t will always take a feasible value, but ∆bit and ∆mit may
take values that violate their constraints, which are 0 ≤ ∆bit ≤ qit and 0 ≤ ∆mit ,
respectively, and y t may also violate its constraints, which are 0 ≤ yt ≤ zt . 43 If there are
any ∆bit < 0 and ∆mit < 0 , these goods are removed from the marginal buyable and
production sets, respectively—that is, the quantity already purchased is held, no previous
purchase plans are revoked but no further purchase plans are made, either—and the agent
proceeds with the next iteration. If there are any i such that ∆bit > qit , the current trading
step is used to exhaust the stocks of the marginal sellers of any such goods, the seller
count for goods i is incremented so as to select the next seller, and the next iteration
begins. If the income plan and the shopping plan are both feasible, optimum has been
achieved and the iteration terminates.
ITERATION 2:
43
There is a notational ambiguity here which I could not think of a way of avoiding. In this paragraph, qit
and zt refer to the capacities of the marginal sellers of good i (to purchase) and the marginal employer (to
hire). Above, the same notation could by implication refer to the buyers and sellers which held rank t in
order of ascending price or descending wage. These will not necessarily or typically refer to the same
retailer. An agent might run through several iterations without incrementing the marginal employer, or the
marginal buyer. I have run up against a paucity of notation, or at least, of notational choices that would be
sufficiently intuitive to be understandable while still giving different entities different names.
251
1.
Start by defining
=
D0 :
{B ; P; d ; d ∀i ∈ B } , where B
0
0
0
i
0
0
∪P =
V , B0 ∩ P =
∅,
d 0 = 1 and di0 = 1∀i ∈ B0 . Then expand B to include any goods i for which pit < 1 , and
create a set Pt consisting of all elements of P except those for which pit < 1 .
2.
On the basis of D t , determine xt, pt, qt, w t , M t , and L t , the parameters for the
optimization of the marginal consumption vector.
3.
Calculate the unconstrained optima of the choice variables, namely, the change
in the money supply ΔM t , marginal market labor y t , the marginal purchases vector ∆bti ,
and the marginal home production vector Δm ti .
4.
If the agent’s income plan proves unfeasible because she exhausts the demand
of the marginal buyer, that is, if yt * > zt , then exhaust the demand of the marginal buyer
and create D t+1 from D t by replacing dt +=
dt + 1 . Return to step 2 and proceed.
1
5.
If the unconstrained optimal marginal consumption for one or more goods is
negative, create D t+1 by redefining Bt +1 :=
Pt +1 :=
6.
{i | i ∈ P , ∆x * > 0} .
t
t
i
{i | i ∈ B , ∆x * > 0} and
t
t
i
Return to step 2 and proceed.
If there are one or more seller stockouts, ∆xit * > qit , create D t+1 from D t by
substituting dit +1= dit + 1∀i | ∆xit * > qit , that is, by stocking out the marginal seller and
selecting a new marginal seller. If for any good i | i ∈ B t , i ∉ P , there is no seller dit + 1 ,
redefine B t+1 to omit good i. If there is any good i | i ∈ B t , i ∈ P, pit < wt , that is, the price
252
charged by seller dit + 1 is greater than the marginal wage, redefine set B t+1 to omit good
i, and redefine set P t+1 to include good i.
7.
If this step is reached, no buyer or seller stockout constraints are binding, and
the optimum has been achieved. End iteration.
That this is a generalization of the solution of the intramarginal problem in the price-taker
case may be seen if we consider what happens when all the qit and y t are sufficiently
large that no stockout constraints are binding. In that case, steps 4 through 6 of Iteration
2 would not be applicable, and there would be only one iteration. Instead, there is a sell
set, which we will call S, and the second stage of the inframarginal problem is to choose
P*(S), the optimal production set for a given sell set. In order to do so, we need ways to
cross-apply the procedures we developed for exploring the production set space in the
price-taking case to the stockout-constrained case. In doing so, we will rely heavily on
defining minima and maxima, e.g., of wages, prices, and labor prices, and for that we
need to define how large is the set of relevant buyers, and of relevant sellers for each
good.
For any nb buyers that we include in the list, we can calculate the quantity of labor,
nb −1
nb −1
∑ y , needed to exhaust the demand of the next-to-last buyer.
j =1
i
If
∑y
=i 1
i
> L − ∑ l0i , this
i∈P
implies that the agent does not have enough labor to exhaust the demand of the next-to-
253
last buyer, therefore there is no use in including the last buyer in the set. This provides us
with rules for choosing nb if Nb sellers are available:
n −1
n
N

 FULL EMPLOYMENT : n | ∑ yi < L ≤ ∑ yi if ∑ yi > L

=i 1
=i 1 =i 1
nb = 
Nb
UNDEREMPLOYMENT : N b if
yi ≤ L
∑

i =1
b
(33)
In equation (33), the concepts of “full employment” and “underemployment” are used to
distinguish between the case where the agent could, if she chose, devote all her time to
paid market work, and case where there is not enough paid market work available to her
to allow this (given her production set). (This is not quite the same as the above
definition.) We can further define the minimum marginal wage wmin = wnb in the “full
employment” case, or wmin = 0 in the “underemployment” case. To determine how many
sellers should be included in the list, we can first calculate the maximum market income
W max , which is:
(34)
Wmax
nb −1
nb −1
nb



FULL
EMPLOYMENT
:
w
y
w
L
y
if
yi ≥ L
+
−

∑ ii =
∑
∑
i
nb 

=i 1
1
=
1
i
i


=
nb
nb

 UNDEREMPLOYMENT : ∑ wi yi if ∑ yi < L
=i 1 =i 1

254
Using W max , and given that the total number of sellers of a good iis N is , we can choose
nis −1
nis
k k
max
i i
=
k 1 =i 1
s
i
n to satisfy either the condition (A)
nis = Nis
∑
i =1
∑p q
<W
< ∑ pik qik , or the condition (B)
pi qi < Wmax . In case (A), the agent could, if she liked, spend her entire income on
good i, and we may say that there is an abundance of good i. Case (B) represents the
case where all traders’ stocks of good i would run out before the agent had spent her
whole potential income on it, or we may say, when the good is in shortage. 44 In case (A),
s
we can define pimax = pini . In case (B), we can define pimax = +∞ .
In the price-taking case, the first condition that was used to narrow down the set of
candidate production sets was the requirement that pi ≥ ws , that is, that the buy price of
good i must be higher than the marginal wage for making good s. This condition can be
adapted to the stockout-constrained case by making a requirement that pi1 ≥ wmin , that is,
that at the minimum marginal wage, it is more expensive (in labor time) to buy good i
than to self-produce it. If this condition does not hold, self-production of good i can be
ruled out.
The other method of culling the production set involved the i  s j relation. This can be
p
redefined as:
44
These terms are used as intuition pumps and do not precisely correspond to the usual use of the terms in
economics.
255
σ
 1 

1
1
 1−σ   pi  σ −1 
1−σ
1
−
−
a
a
 i     j
i S j ⇔ 
  w1  
p



l0i ≤ l0 j

(35)
By replacing
pj
ws
with
p max
j
wmin

max
  pj
1 −  w
  min




σ
σ −1


>0


, we maximize the second term in the first condition in
equation (21), thus making it as hard as possible for i  S j to turn out to be true. If
p
i  S j nonetheless holds, we can be sure that U * ( P ∪ i ) > U * ( P ∪ j ) regardless of how
p
many traders it turns out to be optimal to exhaust. Having adapted i  S j to the stockoutp
constrained case, we can use Iteration 1 without any further modification. And again,
this is a generalization of the price-taking case, since in the price-taking case pi1 = pimax
and w1 = wmin .
The last step is to select the optimal sell set. Consider sell sets S 1 , with wage series
w11 ,..., w1n1 and capacity series z11 ,..., z1n1 , where n 1 is the number of (relevant) buyers of
goods in S 1 , and S 2 , with wage series w12 ,..., wn22 and capacity series z12 ,..., zn22 , where n 2 is
d −1
d
the number of (relevant) buyers of goods in S 2 . Let d1=
( Q ) d | ∑ q1k < Q < ∑ q1k and
=
k 1=
k 1
d −1
d
d 2=
( Q ) d | ∑ qk2 < Q < ∑ qk2 , where Q is an arbitrary quantity of labor; that is, let d 1 and
=
k 1=
k 1
256
d 2 , both functions of Q, refer to the index number of the marginal employer if the agent
chooses Q units of market labor given sell sets S 1 and S 2 , respectively. Sell set S 1 is
preferred to sell set S 2 , and S 2 can be ruled out, if w1d2 (Q ) > wd22 (Q )∀Q < L and
∑l
k∈S1
0k
<
∑l
k∈S2
0k
, that is, if S 1 involves lower overhead than S 2 and the marginal wage
earned under sell set S 1 is higher than the marginal wage earned under sell set S 2 for any
quantity of market labor. Of course, sell set S 1 might consist of a single good, say, good
i, and sell set S 2 of a single unit, good j, and a relation i  j may thus be defined among
s
any two goods. Thenceforward, the procedure followed is analogous to that by which a
production set was selected: any set is chosen as an initial select set; the inverse of the
select set is defined; the outermost set and the core of the select set are defined; the
innermost of the inverse is defined; the union innermost of the inverse with the outermost
of the select set is defined as the candidate set; the core of the select set is combined with
all combinations of goods in the candidate set in succession and utility is calculated for
each of the resulting sets; if any is found which yields a higher utility than the select set it
becomes the new select set and the iteration continues; if not, the iteration is complete.
In short, the solution in the stockout-constrained (general) case consists of four
procedures nested within each other: (a) the optimization of the consumption vector at a
particular margin, (b) the iteration by which the optimal agent decision for a given
production set is identified, (c) the algorithm that identifies the optimal production set for
a given sell set, and (d) the procedure for finding the optimal sell set. The complexity of
257
this algorithm may evoke in some readers a certain skepticism about whether real people
would have the cognitive capacity to make decisions in this way. To this there are four
answers. The first is that because we know by introspection that humans try to behave
rationally, even if they do not always succeed, rationality is an appropriate regulatory
ideal for agents in the model even if the real world is messier. The second is that human
beings’ ordinary, everyday decision-making is really very complex, as becomes clear if
one tries to make a robot imitate it. The third is that what made the above exposition so
difficult was our pursuit of a high degree of generality. We were designing algorithms to
cope with every possible contingency, but real people have the much easier task of
dealing only with the particular situations they happen to confront. Fourth, some of the
major specialization decisions are made only once or a few times in life, and people give
them lots of thought, while for smaller decisions that are made frequently, people can
improve their performance through trial and error.
258
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261
CHAPTER 3.BAYESIAN SKILL REPUTATION SYSTEMS: A CONTRIBUTION TO
‘EPISTONOMICS’
This paper begins with an attempt to understand what economists mean by “knowledge,”
and yields some eccentric yet plausible conclusions about how the macroeconomy works.
A distinctive contribution is the light that it sheds on the issue, important in real
economies but usually neglected by theory, of what it is to “make one’s name.” To make
one’s name is to achieve a success that people talk about, a breakthrough, something that
people identify you by. Ancient Roman generals would actually be named after the
provinces they had conquered: Scipio Africanus was the conqueror of “Africa”
(Carthage). Names can go in the other direction too: companies are often named after
their founders. But even if there is no migration of an actual word from accomplishment
to person or vice versa, a name can become indelibly associated with an achievement:
Bobby Fischer with chess, Babe Ruth with baseball, Bill Gates with Microsoft, Robert
Frost with poetry, Jimi Hendrix with electric-guitar virtuosity, and so on. Or, for
economists, Keynes with depression economics, Hayek with catallaxy, Milton Friedman
with free markets and monetarism, etc. A sign of success is that one can be described
with the definite article: “he’s the guy who…” Rosen (1981)’s work on “the economics
262
of superstars" has a similar theme, but does not explicitly address the issue of fame, i.e.,
how talent is discovered (or not). Of course, many talented specialists are not household
names but are well paid because they enjoy high reputations in a particular field, so the
economics of superstars has a relevance more general than that phrase suggests.
A made name is an economic asset, but what kind of economic asset is it? It will be
referred to here as skill reputation (a phrase also used by Harbaugh (2002)). Skill
reputation does not provide an automatic stream of income, like owning stocks and
bonds. It enables an individual to earn more money by working than he would if he
lacked it or had less of it. But it is not exactly human capital, since it consists not in skills
or abilities on the part of an individual, but in others having information about one’s
skills and abilities. It might be called “social capital,” but that would misrepresent the
conventional meaning of that oft mentioned but vaguely defined concept. Social capital
can mean that a lot of people have affection for you or owe you favors; skill reputation
means solely that people know about your skills and have a high opinion of their quality.
Moreover, skill reputation cannot be a term in a production function (as the word
“capital” suggests), because it it is a factor in the decision to initiate production by hiring
a skill, but does not affect output once production is initiated. (Performance depends on
actual skill quality, not skill reputation.)
In a neoclassical economy with ‘perfect information,’ skill reputation cannot be a scarce
economic asset. Everyone knows you, and what you can do, by assumption. I develop a
263
model which repeals this assumption. I try to convert the magically-endowed perfect
information of the neoclassical model into a kind of market information conceptually
groundedin Bayesian epistemology: agents develop accurate beliefs by updating them in
response to the evidence. The resulting Bayesian skill reputation system is fairly
effective in finding out which skills are good, yet a good deal of wage dispersion
remains. Capital takes on a new meaning in this economy: it does not explicitly consist
in assets that are in themselves productive, though it can be partly interpreted that way,
but serves rather as a fund for paying wages in advance and insuring against losses. The
skill reputation system depends on the economy having a (partial) ‘panopticon’
character, 45 such that once an agent knows about a skill (its own or another agent’s), it
can observe all of the skill’sperformance outcomes, and use them to update its beliefs
about the skill. Inasmuch as agents are able to conceal their failures, the Bayesian skill
reputation system becomes ineffective, capitalist-entrepreneurs are unable to assess risks
properly, and economic efficiency falls off sharply.
Section I reviews relevant literature. Section II presents the model. Section III describes
the results. Section IV concludes.
I. Knowledge in Economics
45
The word ‘panopticon’ is derived from the utilitarian philosopher Jeremy Bentham, who conceived a
prison design in which prisoners could always be watched, and has since become a common metaphor in
the social sciences. Etymologically, it means “all-seeing,” and it can describe systems in which everyone
can see what everyone else is doing.
264
Let me begin with a statement too simple to be quite true, but which will serve as a sort
of outline or mnemonic. When economists talk about knowledge, they generally mean
one of three things:
1.
Designs or ideas, in the sense established by Romer (1990). Designs are
‘excludable’ but ‘non-rival’ goods. Like capital, they can be used in production without
being used up. They are costly to create, and making them is the job of universities,
R&D labs, film studios, etc.
2.
Human capital, knowledge of how to do things, generally acquired through
education and experience, and useful on the job. (Becker, 1962)
3.
Market information, knowledge about availability of goods and about prices,
which in standard neoclassical consumer theory tends to be assumed to be ‘perfect’: the
naivete of ‘perfect information’ is the price paid for the sophistication of neoclassical
consumer theory, which sets up a solvable consumer decision problem by first assuming
that agents are ‘price takers,’ i.e., they are automatically and accurately informed of all
relevant prices.
Now, the first problem here is simply that there are three concepts of knowledge, which
seem quite incommensurable in the role they play in the theory. Are these distinctions
legitimate? Natural language treats knowledge as one phenomenon, not three. Second,
what is the relationship between the three kinds of knowledge? Clearly they are not
independent of each other. Designs are typically of no use unless someone understands
265
how to use them, i.e., designs depend on human capital. Human capital is of no use
unless people know that it exists, i.e., human capital depends on market information.
Often the lines between designs and human capital, or human capital and market
information, are hard to draw. If someone knows the publishing business like the back of
their hand, is that human capital or market information? If I invented a new method and
make my living using it, am I earning my living by human capital or from a design?
Theory does little to elucidate the relationships between these three kinds of knowledge,
and the distinctions between them are ad hoc, motivated more by the needs of particular
theoretical formalizations than by observation or philosophical analysis. 46
To clarify, it might be useful to bring in a concept of knowledge from epistemology, the
subdiscipline within philosophy that specializes in defining what knowledge is. Now,
epistemologists agree that knowledge is, at a minimum, ‘justified true belief’ (Gettier,
1963). Do the three main knowledge concepts in economics meet the (a) justified (b) true
(c) belief standard?
46
A case in point: whereas Romer (1990) attributes long-run economic growth to the accumulation of
designs, Lucas (1988) attributes it to externalities from the accumulation of human capital. Yet what Lucas
has in mind by human capital externalities seems to be the same as what Romer has in mind by designs, for
he writes: “Basic discoveries that immediately become common property – the development of a new
mathematical result say – are human capital in the sense that they arise from resources allocated to such
discovered that could instead have been used to produce current consumption, but to most countries as well
as to most individual agents they appear ‘exogenous’ and would be better modeled as [a change in the
Solow residual].” It seems plausible to consider designs a form of human capital and to regard it as a
misstep in the development of economic theory to treat designs as an independent factor of production, as
Romer (1990) does, since that obscures the fact that designs must typically be known by someone in order
to be used. The real lesson here, however, is that a fundamental re-examination of economic knowledge
concepts is needed.
266
1.
Designs. Designs fail test (c): they are not beliefs. In some cases, they may not
even be in anyone’s head: if a software programmer dies and no one understands how he
designed his best applications, people can still go on using them, and they still enrich
society. More often, someone living does understand the design, but the fact that if many
people understand a design, it is still one design, though there are as many beliefs as there
are believers, shows that design is not belief. It is tempting to say that some kind of
‘collective mind’ of ‘society’ holds these beliefs, but there is no such collective mind, or
at least, if there is, it is supervenient upon individual minds, and how the collective mind
emerges from the interaction of individual minds is something theory ought to explain,
not assume.
2.
Human capital. Some human capital is not belief at all: good looks, physical
strength, sharp eyesight, and immunity to disease are all human capital. Human capital
that is belief is probably more important to the economy than human capital that is not.
But even if we grant human capital test (c), it fails test (b): beliefs, to constitute human
capital, need not be true. A quarterback’s belief that he will win every game may
improve his play, though it is false. Likewise, a salesman’s naïve but sincere belief that
the product of the week is the best thing since sliced bread will make more money than
one whose beliefs are more realistic. If there were no God, a priest might still make a
living by knowing Trinitarian theology, and many professors earn good livings by
teaching theories that (unbeknownst to them) are false. Whether false beliefs are often or
rarely economically useful is difficult to say, but that is not the main point. The point is
rather that even if there is a connection between the truth of beliefs and their economic
267
value, the theory does nothing to elucidate this connection. It does not deal with the
question of truth.
3.
Market information. Market information passes test (c)—an agent’s knowledge
of market conditions is a set of beliefs—and test (b)—the agent in neoclassical consumer
decision theory is always taken to be right about the prices and availabilities of goods—
but it fails test (a): no account of justification is given. True beliefs about market
conditions seem to appear by magic in the agent’s mind. Of course, no theory can
explain everything, but how agents find out about goods and prices in a complex modern
economy is clearly a sufficiently major and important problem that theory ought to have
something to say about it. The whole task of the advertising industry, for example, is to
disseminate such information. Even when the process of search is explicitly considered,
as for example in Mortensen and Pissarides (1994), it consists of an ad hoc matching
function, with no explicit micro level account of how information forms or spreads.
Knowledge is widely considered the single most important factor explaining long-run
economic growth, and is a leading candidate to explain the wealth and poverty of nations.
It is troubling, then, that what economic theory treats as ‘knowledge’ does not meet even
the minimal standard in epistemology for what qualifies as knowledge. To import a more
adequate theory of knowledge into economics is a worthwhile project.
Of course, epistemologists do not agree on what knowledge is, either, and much of what
epistemologists talk about is not suitable for importation into economics. But Bayesian
268
epistemology, as explained in Mark Kaplan’s book Decision Theory as Philosophy
(1996), lends itself particularly well to economic modeling, and I will try to implement
something along those lines here. To translate all three of main economic knowledge
concepts into Bayesian epistemology is too large a task for one paper, so I will focus on
the one which is already closest to philosophical respectability, namely, market
information. Economic theory already conceives of market information as true belief.
All that is needed is an account of justification, of how agents acquire these true beliefs
through Bayesian updating on the basis of new evidence.
Experimental economists have tested whether people behave like perfect Bayesians and
have generally rejected the proposition (see, for example, Charness and Levin, 2003).
However, experiments tend to deal with somewhat special situations. Certainly people
seem to behave roughly like Bayesians. They consider theories more likely to be true
when they observe evidence consistent with the theory. If a theory is probabilistic—if
makes claims not of the form ‘if x, then y, always’ but of the form ‘if x, then y, usually’—
they will consider it less likely that the theory is true when they observe data inconsistent
with the theory (x but not y). If a tremendous amount of data consistent with a theory has
led them to hold it with high confidence, one contrary data point will have hardly any
effect: no one doubts the theory ‘summer is hot’ because of one cold day in July. When
the evidence is scanty, beliefs will be held with less confidence and altered more easily:
“smart girls aren’t as fun to date” is a theory that might be abandoned with one contrary
data point, since a guy could hardly ever have as broad an evidentiary base for such a
269
theory as for the “summer is hot” theory. Bayesian epistemology is used here as a
codification of common sense suitable for implementation by a computer.
B. Animal Spirits and the Winner’s Curse
Now consider the “winner’s curse,” as discussed in Richard Thaler’s book of that title
(Thaler, 1992, Chapter 5). In auctions where people must bid on an item about whose
value they have a noisy signal, e.g., common-value auctions, agents who bid their best
estimate of the item’s value will, on average, take losses, because they are much more
likely to win the auction if they have overestimated the value of the item. Experimental
subjects frequently bid naively and lose money, perhaps because they do not understand
the phenomenon of the winner’s curse. But even if one does understand the theory, it is
very difficult to determine the rational response to it.
It turns out that agents in the Bayesian skill reputation system model running into a
“winner’s curse” problem, and there seems to be no deductive way to deal with it. They
simply do not have enough information to figure out how the winner’s curse problem will
affect their optimal bidding strategy. Keynes arguesin the General Theory that:
A large proportion of our positive activities depend on spontaneous optimism rather than on a
mathematical expectation, whether moral or hedonistic or economic. Most, probably, of our
decisions to do something positive, the full consequences of which will be drawn out over many
days to come, can only be taken as a result of animal spirits – of a spontaneous urge to action
rather than inaction, and not as the outcome of a weighted average of quantitative benefits
270
multiplied by quantitative probabilities. Enterprise only pretends to itself to be mainly actuated by
the statements in its own prospectus, however candid and sincere. Only a little more than an
expedition to the South Pole, is it based on an exact calculation of benefits to come. Thus if the
animal spirits are dimmed and the spontaneous optimism falters, leaving us to depend on nothing
but a mathematical expectation, enterprise will fade and die, though fears of loss may have a basis
no more reasonable than hopes of profit had before… human decisions affecting the future,
whether personal or political or economic, cannot depend on strict mathematical expectation, since
the basis for making such calculations does not exist. (my italics) (Keynes, J.M., General Theory
of Employment, Interest, and Money,Locations 2485-2514)
This passage, and especially the phrase “animal spirits” which has come to allude to and
summarize it, has become famous in part because it goes against the grain of economics,
the social science discipline for which to explain a phenomenon generally means to
explain how it is consistent with rationality. Consequently, later theory in the Keynesian
tradition has generally ignored it, or at most deferred to Keynes by studying the effects of
exogenous changes in investment demand. An economist’s instinct is to regard “animal
spirits” as a placeholder for a better theory that makes the behavior of investors and
entrepreneurs more rational. Yet Keynes’ claim that “a basis for making [optimal]
calculations does not exist” is a good description of the difficulty faced by agents in the
Bayesian skill reputation system in dealing with the winner’s curse.
Clearly, agents should bid less than theirtrue values, but how much less? The answer
depends on knowing a lot of information about other bidders and the distributions of their
signals. The winner’s curse has often been observed in experiments, and Thaler mentions
271
an unpublished paper by Dyer, Kagel, and Levin (1987) in which construction firm
managers, for whom bidding on contracts is an important part of their jobs, participated,
and made the same mistakes in the context of the experiment that they must have learned
to avoid in their professional lives. The authors suggest the following explanation:
We believe that in the field executives have learned a set of situation specific rules of thumb which
permit them to avoid the winner’s curse in the field but could not be applied in the laboratory…
These rules of thumb do not translate into a structurally similar, but different environment which
lacks familiar references… Without theory absorption there is nothing to be carried over from
previous experience. (Thaler, p. 56)
Common-value auctions seem to be a case where behavior “cannot depend on strict
mathematical expectation, since the basis for making such calculations does not exist,”
unless the bidder has, in addition to his own signal, detailed information about the signals
available to other members. Over time, “situation specific rules of thumb” may emerge
in certain areas that enable people to avoid the winner’s curse, but agents may not
understand why these rules work, let alone how to adapt them to changing conditions.
Many have argued that “pessimistic expectations” can sometimes be a cause of poor
macroeconomic performance, but the rise of rational expectations macroeconomics has
made it seem inappropriate to treat expectations as an exogenous causal factor. The
winner’s curse problem suggests that rational analysis of information must be
supplemented by rules of thumb, which may have their own dynamics, such that it might
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be appropriate to regard them for many purposes as an independent causal factor, after
all.
II. The Model
A. Ontology and Structure
The model consists in an interaction of seven types of entities: one World, one Economy,
many Agents, Abstract Skills and Skills, Beliefs, and Opportunities. Structurally, Agents
have Beliefs and Skills, Skills are instances of Abstract Skills, Beliefs are about Skills
and Skills update Beliefs, Skills and their owners (Agents) are hired to exploit
Opportunities, which are generated by the World and allocated to agents by the
Economy.
B. Time and Demographics
The simulation is divided into “years.” Each year some Agents are born, some enter the
work force (at age 20), some retire (with growing probability as they approach age 80),
and some die (with 1/10 probability after retirement). The demographic profile of the
simulation population roughly resembles that of real human populations.
C. Opportunities
Each “year,” Agents receive a set of Opportunities from the economy. Most
Opportunities are risky: they have an upside payoff (always positive) and a downside
payoff (may be positive or negative. If the downside and the upside are the same, the
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Opportunity is not risky, but the quality of the performance is still observable. 47 They
can be exploited only with the help of (a) wealth, and (b) a suitable Skill. Each
Opportunity has a type,and each Abstract Skill has several types to which it is applicable.
D. Wealth
Wealth is needed to pay in advance the owner of the Skill that is hired (even if it is
oneself), and to cover downside risk if the downside is negative. It can be inherited
(Agents choose heirs when they die) but is generally acquired either (a) by exploiting
Opportunities (capital income), or (b) by being hired by others to use one’s Skills (labor
income). Generally, each Agent consumes one-tenth of his wealth at the end of each
round, but this parameter will be varied in Section III.D, to show the effects of the
“savings rate” on economic performance.
E. Skills and Skill Quality
The distinction between Abstract Skills and (instance) Skills is necessary in order to
model the idea that skill quality varies and is not directly observable. All Agents have
general knowledge of what Abstract Skills exist, but they cannot see the quality of any
particular Skill.
Examples may elucidate the concept. Medicine, teaching economics, and cutting hair are
abstract skills. Jill’s medical training, Bob’s ability to teach Macro 200, and Jack’s skill
47
This situation can be interpreted as a task where low-quality work suffices, but high-quality work, though
it contributes no extra value, can be recognized.
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with the clipper and scissors are particular skills, belonging to specific people. Skills are
defined by what they enable those who possess them to do: Jill can diagnose breast
cancer; Bob can make students understand why low interest rates stimulate the economy;
and Jack can turn a shaggy character clean-cut and handsome. These capacities depend
partly on the general class of skills of which particular skills are instances. In general, the
skill of medicine enables a person to diagnose cancer, Jill along with the rest.
However, not all instances of an abstract skill are the same. Maybe Jane is a better doctor
than Jill, and is more likely to diagnose breast cancer correctly. Maybe Robby is a better
barber than Jack, and can produce a more stylish haircut in half the time. The quality of
real skills tends to be continuously-distributed and multidimensional in nature, but in the
model it is binary: all skills are either good, G or bad, ~G. To limit the number of free
parameters, we will assume throughout that each newly generated skill has a 20% chance
of being good. Crucially, the quality of a skill is not observable. Agents cannot directly
observe the quality either of their own Skills or of those of others. They can only make
inferences about skill quality by observing performance.
Each Abstract Skill is defined by the set of Opportunity types to which it is applicable,
and by two success probabilities, one if the skill is good, P(S|G), one if it is bad, P(S|~G).
Each Skill, when used, succeeds or fails with a probability corresponding to its quality.
F. Bayesian Updating of Beliefs
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At the heart of the model is the process by which Agents update their Beliefs about
whether Skills (their own and those of other Agents) are good or bad. This is done by
applying Bayes’ Law, that is, by applying the formula:
(1)
Pt +1 ( G ) P=
=
(G | E )
P ( E | G ) Pt ( G )
P ( E | G ) Pt ( G ) + P ( E |~ G ) (1 − Pt ( G ) )
Where E is an event, P ( E | G ) = P ( S | G ) and P ( E |~ G ) = P ( S |~ G ) if the skilled agent
succeeded in exploiting the opportunity, and P ( E | G ) = 1 − P ( S | G ) and
P ( E |~ G ) = 1 − P ( S |~ G ) if the skilled agent failed to exploit the opportunity.
Of the values in (1), P ( E | G ) and P ( E |~ G ) are properties of the Abstract Skill of
which the hired Skill is an instance. They are known to the Agent as part of general
knowledge. Pt ( G ) is the agent’s prior. The need for priors is arguably a weak point of
Bayesian epistemology, since their initial values must be somewhat arbitrary.
Every Skill “knows” all the Beliefs about itself and updates them whenever it performs.
The interpretation of this is not that the Agent who has the Skill deliberately advertises
his performance: if that were the case, he would advertise only his successful
performances. Rather, all performances by Skills are sufficiently public that those who
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are aware of the Skills see them. Note that this is a rather strong assumption about the
observability of performance.
G. Memory
Beliefs are stored in Agents’ memories. These memories may be thought of as rows of
deep mailboxes, into which new information is pushed in at the front, and out of which it
is pulled out from the front when it is time to use it. Alternatively, they may be thought
of as a “stack,” with new Beliefs put on top, and Beliefs on the top most accessible.
Beliefs are never forgotten, in the sense that Agents always have references to them in
their memories and might in principle recall them. But Beliefs may be effectively
forgotten if they are buried under three or more other Beliefs, though they might still be
recalled and used if Agents who owned all the Skills above those Beliefs in the mailbox
died.
When an Agent has an Opportunity to exploit and is looking for relevant Beliefs, he pulls
out three Beliefs, or all Beliefs that are available, whichever is less, from the front of the
relevant mailbox. If a Skill is used successfully, it is moved to the top of the stack. So
the advantage to a Skill of successful performance lies not only in raising confidence in
its quality, but also in bring it forward in the memory queues.
H. Communication
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Agents are born with Beliefs about all their own Skills and keep these throughout the
game. All other Beliefs are acquired by the following three forms of communication.
•
Casual conversation. Each “year,” an Agent initiates a conversation with one
other Agent and communicates five Beliefs. If the Agent has successfully exploited any
Opportunities in the past year, he communicates these Beliefs about successful Skills
first. For any slots that are leftover to complete the five, the Agent randomly picks a
mailbox and, if there is a Belief in it, adds it to the list. In casual conversation, Agents
have no incentive to misreport, but it is also difficult for them to communicate their
degree of confidence in the quality of Skills. To reflect this, the arithmetic average of the
conversant’s confidence and the default confidence of 20% is taken, after which a
sigmoid function is applied (explained below) to determine the degree of confidence
which the new Agent will invest in the quality of the newly discovered Skill.
•
Customer recommendations. When an Agent has an Opportunity to exploit, he
not only looks up his own relevant Skills, but also calls up two “neighbors” (randomly
selected from the population) and asks them for advice. They look up the Opportunity
type and report the top Skill in their memories, if they have one. The Agent initially
“takes their word for it” and uses these imported Beliefs as if they were his own. But
each one which is not used or which is used but does not succeed is subsequently
forgotten. If a borrowed Skill is used and succeeds, the Agent adopts it as his own, but
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again, only after taking the arithmetic mean of the new Belief’s confidence with the
default confidence level and applying the sigmoid function.
•
Job applications. When any Agent is completely unemployed, he “applies for
jobs” with three other Agents by reporting to them all of his Skills. In this case, however,
since the Agent has an incentive to misreport, the information is completely discounted.
Confidence is set to the default, and the sigmoid function applied.
The way casual conversation and customer recommendations work makes successful
exploitation of Opportunities especially advantageous to holders of Skills. Not only does
the Skill get to update Beliefs about itself favorably, but it also gets itself talked about.
I. Wage Demands
Every non-retired Agent over 20 is both a “worker” and a “capitalist” or “entrepreneur.”
(Agents under 20 are also allocated Opportunities to exploit, though they tend to have
limited wealth with which to exploit them.) This is realistic: real people have both labor
and capital resources.
As a worker, the Agent sets a wage. Between age 20 and retirement the agent has ten
units of labor to dispose of every year. If exactly six of those units are hired, the Agent
asks the same wage the next year. If between seven and nine units are hired, the Agent
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raises the wage by 5% for each unit above six that is hired, and if between one and five
units, the Agent lowers the wage by 5% for each unit less than six that is hired.
If nounits, or all ten units are hired, the change in wage is discontinuous because the
Agent knows that demand for his labor is censored at zero and ten. There, if all ten units
of labor are hired, the Agent raises his wage demand by 20% first, then doubles it, for a
140% increase overall, whereas if none are hired, the Agent lowers his wage by 30% and
then by another 30%, for a 51% decrease overall, or else reduces it to the minimum wage,
whichever is higher. No Agent will work for less than the minimum wage, which may be
interpreted as the minimum income necessary to offset the costs associating with
working, e.g., extra calorie intake needed, commuting, wear and tear on clothing.
These wage evolution rules, though somewhat arbitrary, ensure that Agents whose skills
are in high demand push their wages roughly to the limit of what the marginal employer
will bear, though intramarginal employers may still earn substantial surpluses. Agents’
tendency to push their wages up to levels where they are not fully employed should be
interpreted as a taste for leisure, since they do not pay themselves less, as would be
logical if Agents were trying to exploit market power.
J. The Entrepreneur/Capitalist and the Economic Use of Knowledge
We are now ready to explain the decision process of the Agents in their role as capitalists
or entrepreneurs. The word “capitalist” here needs some elucidation.
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First, “capital” initially meant monies or financial assets that return income, and in
popular usage, it can still mean this. By this account, stocks and bonds are capital. But
economists now use the word to mean physically productive assets like factories. Here
the word “capital” is used in its older, more popular sense. Capital consists of cash in
advance with which to pay wages, and risk capital to cover losses. When an investment
opportunity involves potential negative payoffs (even aside from wages), this can be
interpreted as investments in physical capital which might yield disappointing outputs,
but there is no need to adopt this interpretation explicitly, or to draw a sharp distinction
between capital as a wage fund and capital as a fund to pay for physical apparatus in
advance of production. Capital refers to money, or liquidity.
Second, whereas economics has tended to separate capital and entrepreneurship as
separate factors of production, in this model capitalist and the entrepreneur are the same
function. The entrepreneurial alertness with which an Agent recognizes Opportunities is
in vain if he lacks the capital with which to exploit them. This tended to be true
historically when credit and stock markets were less developed. Today’s more
sophisticated capital markets make it more possible to separate the capitalist and
entrepreneur functions, yet in a sense the model still applies. A capitalist will make the
decision whether to finance a venture based on his understanding of the opportunity.
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The capitalist-entrepreneur’s decision about whether and whom to hire depends on seven
variables:
•
W 0 – the entrepreneur Agent’s wealth. Because the Agent has utility function
U = ln (W ) , his risk aversion falls as he gets richer. Also, if W 0 is too low, the Agent
may be unable to pay wages and cover downside risk, in which case he cannot exploit the
Opportunity.
•
P(G i ) – the subjective probability that Skill i is good.
•
P(S|G) – the probability of success if Skill i is good.
•
P(S|~G) – the probability of success if Skill i is bad.
•
π H – the “high” payoff, received if the Skill successfully exploits the Opportunity.
•
π L – the “low” payoff, received if the Skill is unsuccessful in exploiting the
Opportunity.
•
w i – the wage demand of the Agent who holds the Skill.
Of these, P(S|G) and P(S|~G) are known from general knowledge (of the Abstract Skill),
π H and π L are features of the Opportunity, and w i is a public wage demand. Only P(G i )
is not known with certain and accurate knowledge.
The expected value of hiring each Skill i to exploit an Opportunity, relative to doing
nothing, is:
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( P ( G ) P ( S | G ) + (1 − P ( G ) ) P ( S |~ G ) )U (W + π − w ) +
( P ( G ) (1 − P ( S | G ) ) + (1 − P ( G ) ) (1 − P ( S |~ G ) ) )U (W + π − w )
V (i )
=
(2)
i
0
i
i
H
0
i
i
L
i
−U (W0 )
If max (V ( i ) ) > 0 , the Agent hires Skill i* = arg max V ( i ) , otherwise the Agent ignores
i
the Opportunity.
K. Priors and the Winner’s Curse
Bayesian updating is subject to the problem of arbitrary priors. To mitigate this problem,
a sigmoid function, mentioned above, serves to randomize the value of subjective
probabilities so as to reduce the influence of priors, without destroying strong
information. By definition, a sigmoid function has an infinite domain, but a range of only
(0,1). In this case, the inverse of a sigmoid function is applied, a random number added
or subtracted, and the sigmoid function reapplied. Specifically, from a p base which
depends on how a belief was acquired (see above), the initial prior p initial of a belief is
determined by the following transformation:
(
)
(3)
=
pinitial tanh tanh −1 ( 2 ( pbase − 50% ) ) + u / 2 + 50%, u  ( −1, +1)
The effect of this transformation is to randomize the subjective probability to a
significant extent, while not destroying information if it is sufficiently strong. This is
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consistent with the logic of Bayesian updating. Posteriors generally do not change
sharply vis-à-vis priors when the latter are near 0 or 1. If you are highly confident that a
Skill is good, and see it fail in a task, you will attribute that to bad luck, and adjust your
confidence in the Skill only slightly. Similarly, you will adjust your confidence only
slightly if you see a Skill you were confident was bad perform successfully, attributing
the success to good luck. But if your confidence lay in the broad middle of the spectrum,
you will adjust it substantially in response to any new data.
The sigmoid transformation only prevents implausible unanimity among Agents about the
quality of a Skill; it does nothing to ensure that p base is plausibly chosen. For that, the
default subjective probability of 20% was chosen to correspond to the actual share of
good Skills in the world. However, it turns out that this procedure does not lead to profit
maximization, because of the “winner’s curse.” Agents are more likely to hear about
Skills that succeed. So the Skills the Agent knows about will have been
disproportionately lucky. Furthermore, Agents will tend to hire Skills they have
overestimated. This creates an optimistic bias in Agents’ appraisals of the skills they
hire.
To deal with this problem, we can introduce a pessimism factor λ, to be used in the
following transformation:
(4)
(
) )
(
=
P ' ( Gi ) tanh tanh −1 2 ( P ( Gi ) − 50% ) − λ / 2 + 50%
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Whenever an Agent makes a decision about whether to exploit an Opportunity with Skill
i, it uses P’(G i ) instead of P(G i ). If λ>0, this causes the Agent to invest more
conservatively. Figure 71 shows how the pessimism factor affects profits for certain
parameter values.
Figure 71: A moderate pessimistic bias allows investors to make the most money
So capitalists appear to fare best, for these specifications, with a pessimism factor of just
over 1. Because the optimal pessimism factor varies with the specifications, we will
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generally keep it at zero to avoid unnecessary complexity in the model. Yet Figure 71in
itself represents an interesting finding.
It is plausible that investors must discount their own beliefs and be deliberately
pessimistic when making investment decisions. Consider the position of a venture
capitalist. He is constantly approached by energetic, talented, persuasive people with big
ideas. Yet most of these will fail, and if they do, he is the one who will lose money.
Venture capitalists presumably must deal with this situation by constantly resisting their
impulse to believe, being critical and skeptical, and cross-examining applicants, turning
the plans inside out looking for flaws, and still rejecting a lot of ideas which their best
analytical efforts suggest would succeed. People in capitalist countries, surrounded by
slick advertisements, have to learn a version of the same trick, constantly resisting what
seem, on the surface, like good offers. Note also that while the logic of the winner’s
curse implies that a naïve venture capitalist, or shopper, who believes every applicant or
advertisement, might take losses or at least earn less profit or consumer surplus than a
warier capitalist or show, it does not follow that the applicants and advertisers are liars.
Perfectly truthful advertisements can lead agents to take losses by winning business only
from those who have overestimated the product.
Yet what does it mean to say that a capitalist “believes” a skill is good with 65%
probability, when he intends to act as if he thinks the skill is good with only 20%
probability? It looks as if the capitalist’s beliefs could be summarized, “I believe that the
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Skill is good, but I believe that my belief is false,” which seems like a contradiction. Yet
philosophers like Mark Kaplan (1997) have shown that a naïve translation of high
Bayesian confidence into “belief” is fraught with similar difficulties. For example, in a
fair lottery of 1,000 tickets, we ought to believe, of every ticket, that it will lose, which
implies, if all these beliefs were true, that no ticket would win. In contexts of severe
uncertainty, it might be quite rationalto use quite different Bayesian confidence
investments in one’s own thought and conversation, on the one hand, and in one’s
business decisions, on the other.In fact, Kaplan (1996) routinely assumes that there are
many patterns of investment of confidence in propositions that an agent has not ruled out,
and which can be consistent with the same evidence.
Agents are quite rational in the way they update their Beliefs and make entrepreneurial
decisions, yet there are systematic biases in the subjective probabilities they attach to
their actionable Beliefs, which make them unduly credulous about Skill quality. The
pessimistic bias that has to be introduced to offset the winner’s cursehas to imposed in a
rather ad hoc fashion, and underscores the difficulty of conceptualizing knowledge on a
foundation of rationality. This concept of pessimism is reminiscent of vague market
psychology, of “animal spirits” or “expectations” or “confidence,” and will be interpreted
thus for the rest of the paper. The point is that because Agents (a) have to derive priors
from somewhere, and (b) face systematic biases in the kinds of information they receive,
they succeed best if they can adapt their Beliefs to offset the biases that arise from the
economic system, but to simulate that system and find out analytically what the optimal
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values of those biases are is prohibitively difficult. To that extent, Agents cannot be
rational.
L. The World, the Economy, and the Allocation of Opportunities
The World is the source of Opportunities, and it supplies an array of Opportunities to the
Economy every year, irrespective of the population. Each Agent gets a certain number of
draws, e.g., one hundred. Starting from the first Agent and proceeding to the last (the
order is randomly shuffled every year), each Agent picks a random integer less than the
length of the array, corresponding to some position in it. If that position still has an
Opportunity, the Opportunity is allocated to the Agent and removed from the array. If
the Opportunity has been allocated already the position is empty and the Agent gets
nothing.
When the environment is very rich compared to the population, Agents’ draws will rarely
fall on empty positions, and the Agent will receive nearly one hundred Opportunities
each turn. As the population grows, the environment is depleted and Agents get fewer
Opportunities. On the other hand, there is a larger pool of Skills they can hire in order to
exploit them. We should expect, then, that welfare will initially rise, as Agents get access
to more Skills, and later fall, as the environment is depleted.
The model economy described in this section is the basis for the simulation whose results
are reported in the next.
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III. Results
Having run the simulation many times, with various parameters and modifications, I here
report some of the most striking, intriguing, important and/or suggestive results, first in
summary and then in more detail in the subsections below.
First, the Bayesian skill reputation system which is the distinctive feature of this economy
seems to be fairly effective at creating information about skills. In particular, it almost
never wrongly identifies a high-quality skill. It sometimes fails to recognize high-quality
skills, especially when an agent has other skills that he uses instead. In spite of this, a lot
of wage dispersion persists, even among people with identical skill sets.
Second, the model is able to confirm both the idea that a larger population can deplete the
environment—the carrying capacity concept—and Adam Smith’s notion that the division
of labor raises productivity but is limited by the extent of the market. Carrying capacity
is easy to model, but Smithian growth tends to elude formal modeling. The resource
depletion and Smithian growth effects are prima facie inconsistent, since the former
implies that population is good for growth, the latter that it is bad for it. The two effects
offset each other, their relative strength varying with the quantity and nature of
opportunities and the variety and distribution of skills, so that a greater population may
either raise or lower living standards. If opportunities rise proportionally with
population, the Smithian growth effect is isolated and living standards rise.
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Third, though capital in this model is not a factor of production in the usual sense, it is
necessary, scarce, and earns (on average) profits, which are the rewards of the capitalistentrepreneur for (a) the patience to accumulate reserves, (b) the alertness to “notice”
opportunities, and (c) taking risks. I show that one stylized fact about real economies
also appears in the simulation, namely, that capital income is more volatile than labor
income. I also show that a rise in the “savings rate”—a fall in the share of wealth that
agents consume each round—leads to a rise in GDP per capita. In that sense, even
though“capital” is mere liquidity instead of productive assets, it affects output as if it
were a true factor of production.
Fourth, as already suggested, the “pessimism factor” lends itself to an intruiging
interpretation as Keynesian animal spirits. When pessimism is increased exogenously, it
has a depressing effect on the economy similar to that which Keynes thought pessimistic
expectations would have. It serves as a device to coordinatequasi-collusion among
capitalists, and raises capital’s share of income, but it also depresses the economy,
leaving capitalists with slightly less total income, while labor’s income falls sharply and
wage inequality increases.
Finally, I explore the role of the most distinctive feature of the model, the Bayesian skill
reputation system itself. That the skill reputation system turns out to be absolutely
essential for the economy is shown by an experiment in which agents suddenly lose all
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their information about the existence and quality of one another’s skills. The result is
catastrophic: GDP per capita falls to almost nothing. Yet recovery occurs very swiftly,
within just a few rounds. My intuition that it might take the economy a long time to
recover from a negative shock to the quality of its information was not confirmed under
any of the specifications that I tried.
But great damage is done if agents are allowed to conceal failures, thus vitiating the
information about skills that is generated by the skill reputation system: even when the
“pessimism factor” is endogenous, capitalists are not able to adapt to this change, and
they get fooled, take bad risks, and lose money.
A. The Economy Exhibits a Steady State
Before looking at the model in detail, I must reassure readers by showing that the model
tends towards a rough equilibrium or steady state. Without some kind of equilibrium, it
is difficult to know what reports to result, since the model might be in many different
states.
Figure 72shows that, after a transient “initialization,” the model macroeconomy exhibits
just such a steady state. In the steady state, capital does not accumulate, because the
fixed proportion of their capital which agents consume every turn is just equal, on
average, to what they are able to earn through labor and entrepreneurship. New
generations succeed old ones, but each draws its skill sets from the same distribution, and
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the law of large numbers ensures that differences in talent do not vary too much over
time. The distribution of exogenous opportunities is the same.
Figure 72: The economy converges to a steady state in which GDP per captia is nearly constant over
time
Some cyclical processes seem to be at work in Figure 72, but they are not our focus.
Rather, having established that a rough steady state exists, we will explore what is going
on inside the simulated economy. Since the steady state reliably emerges after a short
initialization, the practice throughout this section will be to initialize the economy before
deriving results to report.
292
B. Making One’s Name: Reputation and the Distribution of Wages
A stylized fact about real world economies is that they exhibit highly skewed and unequal
wage distributions, even within professions, such as have been documented by Piketty
and Saez (2006), Card and DiNardo (2002), Katz and Autor (1998), and Rosen (1981).
The title of Rosen (1981), “The Economics of Superstars,” might suggest that he offers a
model of fame, since superstars, of course, are famous, but in fact Rosen is working in a
neoclassical world of perfect information. By contrast, in the context of a Bayesian skill
reputation system, fame does have a well-defined meaning.
The simulated economy exhibits a high degree of wage inequality within industries. In
this respect, it captures a feature ofreal economies (especially in recent decades) which
“representative agent” models have difficulty capturing. Figure 73represents one of the
industries (chosen at random) from the same economy represented in Figure 72, with
wages on the horizontal axis, and reputation quality or average expected probability of
=
success, that is, the average
of P ( S ) P ( G ) P ( S | G ) + (1 − P ( G ) ) P ( S |~ G ) for all
agents who have beliefs about the quality of each agent’s skills, on the vertical axis. The
size of the bubbles shows how many times the skill was hired.
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Figure 73: The distribution of wages among those working in a profession is highly skewed
The most striking feature of Figure 73is the high degree of wage dispersion. Among
agents who were hired in this industry, wages range from 1 (the minimum) to 298. Also,
wages are clearly correlated with an agent’s reputation quality.
However, the relationship between reputation quality and wages is not linear. Agents
with reputation quality less than 80% usually make less than 20 and always less than 50,
and only a small rise in wages is visible as reputation quality rises from 40% to 80%.
The super high earners all have reputation quality well over 80%. One agent with
reputation quality of 93% and a wage of only 15.5 is visible in the upper left corner. This
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agent is a bit of an outlier from the pattern. The market knows his services are a bargain,
and hires him for six units of labor, more than any other agent in the industry.
Finally, there are some skills that have high quality reputations and demand high wages,
yet were not hired in this industry. The reason for this is that each skill can be applied to
several different opportunity types.
Figure 74shows wages against the other dimension of reputation, reputation reach. The
vertical axis shows how many beliefs are extant about a particular skill, while, as before,
the horizontal axis shows the wage, and the size of the bubble shows how many times the
skill was hired.
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Figure 74: Highly paid specialists all enjoy far-reaching reputations
Reputation reach is clearly correlated with the wage. Again, though, the relationship is
not linear. All of the skills that earn high wages, without exception, are fairly well
known, with at least 100 extant beliefs about them. But the converse is not true. Some of
the skills that are well known do not earn high wages. This may be (a) because although
they are widely known, they have poor quality reputations, or (b) because they have only
just become well known and the market has not had time to drive up their wages yet.
And again, some skills are well-known, demand high wages, and are not hired.
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Figure 75is designed to show both dimensions of reputation in the same chart. On the
horizontal axis is reputation quality, on the vertical axis, reputation reach. This time, the
size of the bubbles represents the wage, and the color, transparency, and style of the
bubbles indicates (a) whether the skills were hired or not, and (b) whether they are good
or bad quality.
Figure 75: Only skills that are truly of good quality acquire good reputations
297
The main result to be noted in Figure 75 is that all the skills that enjoy high reputations,
that is, that enjoy reputation quality over 80%, are of good quality. There are many
“false negatives,” skills that are actually of good quality but have not been recognized as
such by the markets, but there are no “false positives,” skills that are really of bad quality
but are widely held to be of good quality.
Among good skills, reputation and reach are correlated. Causation runs both ways here.
Skills that are known to be good by a few people will be hired by them, and success will
make them better known. And if a skill is widely known, it will be hired frequently and
will have a chance to prove its quality.
High wages are an obstacle to some skills’ being recognized. If an agent has more than
one skill of good quality, success in one field may lead the agent to raise its wage
demands, and thus prevent its skills in other fields, which do not yet enjoy high
reputations, from being hired and proving themselves.
When large disparities in wages are observed among people in the same profession, an
obvious explanation is that there are differences of talent among professionals with the
same specialization. Such differences of talent, though, are generally harder for the
researcher to observe than for market participants. In the skill discovery simulations, by
contrast, the modeler (me) enjoys privileged access to the truth about skill quality which
the market has to learn, imperfectly, by induction. Figure 76takes advantage of this, and
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shows that even when agents with the same skill level are compared, a large amount of
wage dispersion remains.
Figure 76: There is persistent wage dispersion, which appears to be largely unrelated to age
Each point in Figure 76 represents (not a simulation run but) a single agent, the vertical
dimension showing his wage, the horizontal dimension, his age. For the sake of
establishing as clearly as possible the relationship between wages and skill sets, the
specifications of the model are not the same as in the earlier Figures, but have been
chosen so that all of the agents in Figure 76have exactly the same skill set: one skill, of
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good quality, applicable to opportunities of type 0, where the abstract skill has an 86%
chance of success if the skill is good, 46% if the skill is bad.
Since these agents can provide the same service, the naïve neoclassical view would hold
that they should earn the same wage. In fact, the highest earners among agents of this
type earn four to five times as much as the lowest earners. Yet the wages of agents with
the high-quality version of this skill, even at the low end, are higher than the wages of
agents with the low-quality version of this skill, whose wages are generally below $50
and almost never exceed $100.
The causes of this wage dispersion are difficult to determine and would probably require
a deeper analysis of network effects than I am able to undertake. Figure 77shows the
correlation of labor income (not wages), with reputation reach. There is a small but
statistically significant correlation between labor income and reputation reach; in this
case, each other agent who knows about a skill raises the labor income of the skill’s
owner by about $1. But less than 1% of the variation in labor income is thus explained.
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Figure 77: Reputation reach is associated with higher labor income, but not very strongly
Some of the differences in labor income are doubtless explained by mere chance, i.e.,
whether other agents who are aware of a particular agent’s skill happened to come across
relevant opportunities. Also, not only the number who know about an agent’s skills, but
the wealth of these agents, and their ability to exploit relevant opportunities when they
arise, are probably among the determinants of agents’ labor income. In the absence of
perfect information, it is not surprising that the “law of one price” does not hold even
among agents who have exactly the same skill set, but the degree of wage dispersion in
the model remains puzzling, yet on the other hand, realistic.
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It is worth noting, too, that Figure 76 seems to show at most a slightly rising age-earnings
profile. A skill discovery narrative could in principle rival the human capital narrative in
explaining the widely observed empirical fact that earnings tend to rise with age. By this
account, it is not that older people are actually more skilled than young people; rather, the
market knows more about their skills.
There is a parallel between the contrasting interpretations of wage distributions offered
here and in Rosen (1981), and the debate between the human capital school in the
economics of education, and the signaling school (Weiss, 1995). Rosen (1981) assumes
that talent is observable, and attributes differences in earnings to differences in talent.
This model suggests that there might be large wage dispersion among people who have
the same level of talent, because of differences in information about talent. Not talent
alone, but a made name is what leads to high earnings.
C. The Effects of Population: Resource Depletion versus Smithian Growth
There are at least two different effects that population can have on living standards: (a) it
might raise them, by allowing a more refined division of labor which leads to higher
productivity, or (b) it might lower them, by depleting natural resources that are inherently
scarce. I will call the first of these the Smithian growth effect and the second the resource
depletion effect.
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The scarcity of natural resources is an inherent feature of the model, since a finite number
of opportunities is supplied by the natural world. But since there is also a need for varied
skills in order to exploit opportunities, a tendency towards Smithian growth is also builtin. Figure 78shows how these two effects offset one another as the population rises.
Figure 78: GDP per capita eventually falls with higher population, but it initially rises
InFigure 78, the horizontal axis represents population, the vertical axis, GDP per capita.
There are 5,000 opportunities of 100 different types, drawn from the same distribution of
opportunities, and agents draw their skill sets from the same distribution of skills. Each
simulation run is initialized for 200 turns, and then an average of GDP per capita is taken
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over 100 turns. 48 When the population is greater than 200, GDP per capita declines as a
larger population depletes the available opportunities.
We can isolate the Smithian growth effect by varying the number of economic
opportunities proportionally with the population, instead of holding it constant. The
result of this approach is shown inFigure 79.
Figure 79: A larger population raises productivity through a finer division of labor
48
The technology has been designed to include more opportunity types and a few very high-risk
opportunities, compared with the technology underlying the economies displayed in some of the early
Figures.Some technology specifications work better than others for highlighting particular results, but the
general pattern of results will arise for wide ranges of specifications.
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Each point in Figure 79 represents a simulation run. The birthrate is chosen, and the
number of opportunities is proportional to the birth rate, ranging from about 200 to about
10,000. The number of opportunities per member of the population is roughly constant
(not precisely constant since there is a random element in the birth, retirement, and death
rates) across all runs.
The result is clear. A larger population leads to higher GDP. So as Adam Smith said,
“improvement in the productive powers of labor… [is] the effect of the division of labor”
and “the division of labor is limited by the extent of the market.”
D. Factor Shares, Reconceived: Capital and Labor Incomes in the Skill Discovery
Economy
How is the social product allocated between capital and labor? And what is capital,
anyway? The traditional answer to these questions begins with a production function,
e.g., Y = K α L1−α . Typically, the production function has constant returns, because
anything else creates problems for the traditional theory of competitive markets. Second,
it is assumed that markets are competitive, so each factor earns its marginal product. In
the Cobb-Douglas case, w=
(1 − α ) K α L−α and
r = α K α −1 L1−α . Total capital income is the
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product of capital and the interest =
rate, e.g., rK K=
(α K α −1L1−α ) αY . Total labor
income is the product of labor and the wage, e.g., wL =
L ( (1 − α ) K α L−α ) =
(1 − α ) Y .
The Cobb-Douglas case yields a very neat result. The sum of labor and capital income is
precisely Y, confirming that the market equilibrium is feasible, and the Cobb-Douglas
exponent on capital α can be interpreted as capital’s share of income. Since, empirically,
capital’s share of income is about ⅓, it is concluded that the “real” aggregate production
function is approximately Y = K 1/3 L2/3 . In what sense this can be true is very hard to say.
Particular production processes do not tend to have this kind of functional form, and the
very concept of aggregating the highly diverse entities that are included in the category
“capital,” measuring them all in dollar terms although their relative prices regularly
fluctuate, is problematic. This was the theme of the “Cambridge capital controversy”
(Cohen and Harcourt, 2003), in the course of which Piero Sraffa, Joan Robinson and
others made telling criticisms of the production function concept which were ultimately
marginalizedbecause these criticslacked a convenient alternative way to study the issues.
In this model, “capital” is a fund to cover wages in advance and to insure against losses.
When an agent hires another agent (or his skill) to exploit an opportunity, taking on
himself the risk, he is behaving like a capitalist, or entrepreneur: in the context of this
model, these two functions are combined. Without “capital,” an agent cannot exploit
opportunities, but nor can capital generate income unless opportunities appear. Capital
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income may be interpreted both as (a) the returns to patience and delayed consumption,
and (b) the rewards for the entrepreneurial alertness to notice opportunities.
Figure 80tracks capital and labor income over 500 turns of a simulation run.
Figure 80: The raw data on labor and capital income
This raw data is hard to interpret visually, but a few things can be noticed. First, capital
and labor income seem to be roughly the same, implying that capital’s share of income is
approximately ½ in this model (and with these technological specifications). By and
large, agents earn about the same amount in their roles as workers, doing jobs for others
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in return for a wage, and in their roles as capitalist-entrepreneurs, hiring others to use
specialized skills to exploit opportunities which the agent has the unique advantage of
having noticed. Second, both series exhibit volatility. Third, the series seem roughly to
move together. The last observation is clearly confirmed inFigure 81.
Figure 81: Capital and labor income tend to rise and fall together
An OLS regression shows that the correlation between capital and labor income is highly
significant.
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To analyze the relative volatility of capital and labor income, Figure 82shows the
difference between labor and capital income and a 5-turn moving average of each of
these variables.
Figure 82: Capital income is more volatile in the very short run than labor income
In the short run, that is, in any given “year,” capital income exhibits much larger
deviations than labor income. Swings of $30 per capita, i.e., of 15% or more, in the
capital income deviation, are quite common, and it is typical for the absolute value of the
capital income deviation to be more than $10. By contrast, the labor income deviation
very rarely exceeds $10 and is usually less than $5.
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Why is capital income more volatile? The reason seems to be the combination of pricestickiness and residual-claimancy. Wages in this model are not always sticky: agents
may double or halve their wage demands in a given round. Still, at any given time the
wage is a legacy of the past, and there are limits to how quickly it can move. The
entrepreneur-capitalist’s income is comprised of whatever is left over. His income bears
the brunt of the volatility of the opportunities available in the environment itself, while
the worker is somewhat insulated from this.
A peculiar feature of the perfectly competitive economy with a Cobb-Douglas aggregate
production function is that capital’s share of income is a constant, regardless of the
endowments of capital and labor. No such relationship holds in the model presented
here. Far from having any tendency to be constant, the capital share will shift in response
to many variables. For example, Figure 83shows how the capital share of income
changes as the environment becomes richer in opportunities.
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Figure 83: The more opportunities there are in the economy, the larger is capital's share of income
Figure 83 shows results from 300 simulation runs, in which everything is kept the same
except for the number of opportunities that appear in the economy per year. The
horizontal axis shows the number of opportunities (a parameter), the vertical axis,
capital’s share of income, and each point represents a 100-turn average of capital’s share
of income after 200 turns of initialization, against the number of opportunities for a
particular simulation run.
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When the number of opportunities is less than 3,000, this economy is stuck in a poverty
trap. Chances to discover skills and earn easy capital to finance larger ventures are too
few and far between. This means that workers are so underemployed that their wage
demands are rock bottom, and almost all the rewards from the few opportunities that do
get exploited go to capital. So this shows up as a capital share of income near 100%.
At a little under 4,000 opportunities, a critical mass is reached, a significant amount of
production begins to occur, and wages are bid up. As a result, the capital share falls
(though absolute capital income, of course, is much higher). Capital share falls
continuously, though with decreasing steepness, as the number of opportunities rises to
50,000 opportunities. By this account, “you need money to make money” is truer of
poor, opportunity-scarce economies than of rich ones.
“You need money to make money” is true not only at the level of individuals, but of
society as well. Economists have long held that high savings rates lead to capital
accumulation and to higher GDP. Figure 84 shows that the same pattern holds true in the
skill discovery economy.
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Figure 84: When agents consume more of their wealth each turn, GDP is lower
In Figure 84, the horizontal axis shows the consumption rate, the vertical axis, GDP per
capita, and each point, the average over 90 turns of GDP per capita after an economy is
initialized with an 80% consumption rate and then experiences an exogenous change in
the consumption rate. When agents consume 90% of their wealth each turn, GDP per
capita is about 100. When agents consume 60% or less, GDP per capita is upwards of
250.
That higher savings lead to higher GDP per capita is a stylized result of the neoclassical
production functionmodel. However, it holds here for adifferent reasons. It is not that
capital per se is a productive asset, but rather that having capital enables agents to exploit
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profitable opportunities when they arise, and makes them more willing to take risks.
More risk-taking entrepreneurs implies more opportunities for agents to prove their skills,
and this skill discovery information feeds back into higher economic performance. It
appears that capital as mere liquidity can function, from the aggregate perspective, as a
term in the production function.
E. Animal Spirits as Employer Pessimism
As we have seen, an “employer pessimism factor” is introduced into the model as a
means for agents, in their role as entrepreneur-capitalists, to overcome the winner’s curse.
It seems clear that there can be no rational, deductive method for determining what the
pessimism factor should be at any given time. As already discussed, I try to make a
virtue of the ad hoc nature of the pessimism factor by interpreting it as Keynesian animal
spirits.
Altering the pessimism factor exogenously affects both GDP and capital’s share of
income, as shown inFigure 85 and Figure 86.
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Figure 85: Pessimism on the part of capitalists reduces GDP per capita
In Figure 85, each point represents a simulation run, with its horizontal location
representing the pessimism factor by which capitalists adjust their simple expectations
when making investment decisions, while the vertical location represents the average
GDP per capita in the simulation economy over 100 turns, after initialization. The trend
is clearly downward, implying that when capitalists are more pessimistic, GDP per capita
is lower. The economy does best when capitalists are optimistic and aggressive.
However, this results in a lower capital share of income.
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Figure 86: A climate of pessimistic expectations acts as a form of collusion, raising capital's share of
income
Even as they reduce GDP per capita, pessimistic attitudes lead to a higher share of
income for capital. In effect, pessimistic expectations act as a substitute for collusion. A
worker cannot demand higher wages because all his potential bosses are wary of
investing, and need to foresee a larger chance of profit before they will take action. Since
capital gets a larger share of a smaller GDP per capita, the net effect is ambiguous. In
this case, though, GDP per capita falls more steeply than capital share of income rises, so
pessimism reduces total capital income, even when it is advantageous for individual
capitalists, up to a point.
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An optimistic capitalist creates more value than the profits he earns, since he also pays
wages to workers. While wages are partly compensation for the disutility of working,
workers tend to enjoy a surplus from their participation in the labor market, so that by
creating jobs and lifting the prevailing wage rate, the capitalist generates positive
externalities. Even if the capitalist himself loses money at the margin, because the wages
he pays are greater than the average revenue he takes in, the investment may benefit
society.
We can take a closer look at the effect of changes in the pessimism factor by running an
“experiment” in which the factor is exogenously altered. Figure 87 shows ten runs of the
simulation with all parameters held constant, except that during each simulation run, the
pessimism factor is initially 1.0, then, after the economy has settled into a steady state,
pessimism jumps to 5.0. That is, entrepreneur-capitalists start to be much more cautious.
This results in a drop in GDP, most of which occurs almost immediately, while some of it
is spread out over the next thirty or forty turns.
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Figure 87: A rise in pessimism (a decline in "animal spirits") leads to a drop in GDP per capita
The surge in pessimism also results in an increase in capital’s share of income, as shown
inFigure 88.
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Figure 88: The rise in pessimism raises the capital share of income by about 10%
Even though capital’s share of income rises, because GDP per capita fell, the
entrepreneur-capitalists still earn less money overall than they did before the rise in
pessimism, as shown inFigure 89.
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Figure 89: Capital income falls a bit when pessimism rises
Some of these patterns are reminiscent of certain stylized facts about recessions and
depressions. For example, in the United States since 2008, 49 GDP has been well below
trend, wages have stayed low while corporate profits have been relatively healthy, but
holders of capital have done worse than they did in the preceding boom. However, the
next result is not intuitively especially associated with recessions and depressions. A rise
in pessimism leads to a sharp increase in wage inequality, as shown inFigure 90.
49
This was written in summer 2011.
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Figure 90: Pessimism leads to a rise in wage inequality
The reason for the rise in wage inequality shown in Figure 90is easily seen if we consider
the determinants of wages and how the pessimism factor operates. With a high
pessimism factor, entrepreneur-capitalists only trust workers for whom the subjective
probability that they are good quality is very high. More workers get stuck in “foolproof”
jobs, bidding the wages of these jobs down, while barriers to entry for, so to speak,
“professional” jobs, in which individuals are employed because they are trusted to do a
good job, become higher.
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It seems unlikely that pessimism about worker quality is a major factor in recessions and
depressions. But it is plausible that recessions and depressions are driven by the way
business respondent to the winner’s curse. It might be that the most rational methods of
decision-making, given the incomplete information available to economic decisionmakers and especially to investors, would lead to systematic errors, and agents evolve ad
hoc rules of thumb to cope with it. Such rules of thumb are not robust to changing
conditions, and may also lead to systemic crises (e.g., if banks get in the habit of
gradually increasing leverage).
But of course, if I want to offer a theory of business cycles based on investors’ responses
to the winner’s curse, it will not do to vary the pessimism factor exogenously. It would
be necessary to explain why the factor shifts, i.g., to make it endogenous. The following
method which attempts to do so was implemented. Agents are assigned pessimism
factors λ *t + µ , where μ is uniformly distributed between -1 and +1. λ *t is initially set
to 1. Thereafter, it is chosen each turn by running a regression π i =β 0 + β1λi + β 2 λi 2 + ε ,
where λ i is the pessimism factor of agent i, π i is the capital income of agent i, β 0 is a
constant term, β 1 and β 2 are regression coefficients, and ε is an error term, then
λ *t +1 if pt − λ *t >10
− β1

*t +1 0.9λ *t +0.1 pt if − 10 < λ *t − pt <10 is
calculating p* =
. After that, λ=
2β 2
λ * −1 if λ * − p >10
t
t
 t
calculated, and becomes the basis for the next distribution of pessimism factors. The idea
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is that agents’ pessimism factors are centered around the optimal value, as estimated
based on agents’ experience, but values are randomized for exploratory purposes.
This algorithm is moderately successful in maintaining the pessimism factor in the
vicinity of the optimal level. At any rate, the factor does not follow a random walk: it
clearly has a central tendency. And the central point to which it tends is within a
reasonable range, averaging about one, enough to compensate for the winner’s curse
without inducing too much neglect of profitable opportunities. The factor also exhibits
large and apparently persistent variations, drifting up and down and remaining at high or
low levels for prolonged periods of time. Figure 91 shows the time paths of the
pessimism factor for ten simulation runs over 1,000 turns, after initialization.
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Figure 91: Pessimism, arising endogenously, has a clear central tendency
Not surprisingly, GDP also has a central tendency in the endogenous pessimism case. It
exhibits no strong correlation with the pessimism factor, however, a result somewhat in
contrast with the finding that when pessimism is exogenously varied, higher pessimism
tends to reduceGDP. Many other algorithms might be designed to drive the evolution of
responses to the winner’s curse in labor markets, so possibly another algorithm would
bear more resemblance to the stylized facts of business cycles.
F. What is the Contribution of the Skill Reputation System to the Economy?
The Bayesian skill reputation system is the most distinctive element in the present model.
It is therefore of interest to know what the contribution of information about skills is to
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the performance of the economy. One way to answer this question is to intervene in the
simulation by causing information about skills to disappear, then see what happens to
GDP. Figure 92 shows what happens when all the agents’ memories are erased.
Figure 92: When all skill information is suddenly lost, GDP per capita plunges, but recovery is swift
Clearly, information about skills is essential to the economy. Without it, GDP per capita
falls to near zero. On the other hand, the economy recovers with startling swiftness from
the catastrophe. Within about five turns, GDP per capita has returned to its pre-
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catastrophe levels. The swiftness of the recovery is indicative of the efficiency of the
Bayesian skill reputation system.
To ask how much skill reputations contribute to the economy turns out to be a bit like
asking how much labor or physical capital contribute to the economy in the neoclassical
model. If Y = K α L1−α , there can be no production without both labor and capital.
Likewise, in the present model, skill reputations are an indispensable “factor of
production.” On the other hand, the ease with which skill reputations can be replaced
when destroyed sets them apart from capital in the neoclassical model, which has to be
laboriously saved over time.
I did not find model specifications which would support the intuition that it should take a
long time for the economy to rebuild information about skills if this were somehow lost.
G. What Happens if Agents Have Some Control over Information about Themselves, and
Can Conceal Failure?
The neoclassical model has a ‘panopticon’ character: everyone can see what everyone
else does. In the present model, agents do not have perfect information, but once an
agent knows about a skill, he is automatically able to see everything that skill does. This
is quite a strong assumption, and we will now relax it.
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Economic activity depends on information, and the Bayesian skill reputation can provide
it, but when agents are able to conceal their failures, the skill reputation system becomes
ineffective. Of course, when an agent performs badly, while it is in the interest of all the
entrepreneur-capitalists who might consider employing him to know about this, it is in
the agent’s interest to mask it. Up till now, then, we have been tacitly assuming, either
that the agents are too honest to mask their poor performances, or else that they are
unable to do so.
I will now introduce the alternative assumption that agents can try to mask their bad
performance, and succeed in doing so with probability p. The analysis hitherto all
concerns the special case of p=0. I now allow p to vary between 0 and 1 for different
runs of the model. This modification is, of course, realistic. On resumes, people usually
try to highlight their successes, and omit their failures. They get recommendations from
people for whom they have done good work, not from those whom they alienated, or for
whom they shirked or made damaging mistakes, though potential employers might be
even more interested in a candidate’s bad work than in his good work. Of course, this
selective concealment will create a bias in Bayesian updating.
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Figure 93: GDP per capita falls by about 20% when agents learn to conceal their failures with 50%
probability
Figure 93 shows ten runs of an experiment in which agents are initially unable to conceal
their failures, and then become able to do so with 50% probability. That is, if an agent
succeeds, the success is visible to all of the agents who have beliefs about that skill, but if
an agent fails at a task, each of the beliefs that refer to that skill may or may not be
updated. The result is a swift fall in GDP and a new, lower steady state, subject to
fluctuations that are of about the same magnitude as before.
I was surprised to see that labor income does not drop at all when the opportunity to
conceal failure arises, as shown inFigure 94.
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Figure 94: Labor income stays at exactly the same level before and after the opportunity for failure
concealment appears
Instead, all the losses under the new system are borne by capital income, as shown in
Figure 95.
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Figure 95: Capital income drops sharply when agents learn to conceal failure
The reason that I was not expecting this result is that the algorithm for endogenizing
pessimism, introduced in Section E, was operating. Figure 96shows that the prevailing
pessimism factor rose slightly after failure concealment was introduced.
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Figure 96: When agents are able to conceal their failures, employers become slightly more pessimistic
While employers do become more pessimistic about agents when failure concealment
becomes possible, the rise is only a slight one, and it is inadequate to offset the
deterioration in the quality of information available to employers and enable them to
maintain profits.
The next four Figures establish these results far more clearly, by showing the variations
in (a) GDP per capita, (b) labor income, (c) capital income, and (d) pessimism as the
“visibility of failure,” i.e., the probability that when an agent is hired to do a task and
fails, this failure is visible to any given belief that refers to the skill used. Five hundred
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runs of the simulation are shown, and each point represents an average value over 200
turns after initialization. GDP per capita rises with the visibility of failure, from around
50 when the visibility of failure is zero, to between 300 and 350 when the visibility of
failure is 100%.
Figure 97: GDP per capita is an increasing function of the information available to agents
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Labor income turns out not to be completely independent of the visibility of failure. It is
unaffected by failure concealment as long as the visibility of failure is 30-40% or more,
but when visibility of failure is below this, labor income begins to fall. At this point,
capital is so depleted by bad risk taking that employers are short of funds to hire agents.
Figure 98: Labor income is constant for visibility of failure over 30-40%, but falls when failure
concealment depletes capital
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Figure 99 shows clearly that capital income collapses as agents learn to conceal failure.
Entrepreneurs have no effective defense against agents who report their experience
selectively. As visibility of failures falls, they get fooled about workers’ skills and start
losing money.
Figure 99: Capital income falls off as failures become less visible
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Finally, Figure 100 shows that while employers initially do become more pessimistic
about workers as the latter begin to conceal failures, the response of pessimism to failure
concealment is rather weak. Below 40%, the algorithm to endogenize pessimism starts to
break down, and pessimism outcomes become scattered and patternless.
Figure 100: Employer pessimism responds weakly to declining visibility of failure, and then becomes
confused
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To sum up, the failure concealment experiment reveals the limitations of the Bayesian
skill reputation system for generating market information. While the “pessimism factor”
seems to be adequate (not necessarily optimal) for dealing with the winner’s curse
problem, when agents begin to conceal their failures, employers are not able to make
even approximately accurate influences about agent capabilities, and start to make
systematic mistakes and lose money. Since, of course, failure concealment is a pervasive
feature of real world economies—product markets as well as labor markets—some richer
theory of knowledge is probably needed to deal with it.
While the experiment varies the failure concealment as an exogenous parameter,
employers may actually be able to influence how effectively job candidates are able to
conceal negative information by choosing their recruiting style. Granovetter (1983)
showed empirically that networks of weak ties are very important for finding jobs. The
model suggests one explanation of this. Job candidates who respond to an advertisement
are in a good position to control the information that reaches the employer. It is easy to
leave unflattering facts off of a resume, and to choose people to write recommendations
whom one knows will provide a favorable rating. By contrast, a third party who knows
both the employer and the candidate, and who may care about the employer’s interests
and/or be concerned about his own reputation, will be more likely to report negative
information. Even a weak tie—a contact who actually knows quite little about the
candidate—may serve to solve the failure concealment problem, as a mere resume does
not. The experiment may also explain why (according to folk wisdom)employers do not
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like gaps in a resume: by insisting on knowing what a candidate was up to at every
period in his life, a prospective employer limits the candidate’s opportunity to conceal
failure.
IV. Conclusion
The analysis offered here is intended as a down payment on a much more ambitious
theory, a comprehensive ‘epistonomics’ that defines economic knowledge and explains
the use of knowledge in society. This would probably turn out to be a theory with one
foot in epistemology, another foot in economics, and a third foot in public opinion and
even politics: it might explain revolutions as easily as stock market bubbles.
What has been established so far is that in an appropriately structured model, Bayesian
learning can be a fairly good substitute for the perfect information assumed in traditional
neoclassical models. In spite of the impressive efficiency of the Bayesian skill reputation
system, a high degree of wage dispersion persists. The model suggests explanations of
some stylized facts in macroeconomics, such as greater volatility of capital income
compared to labor income, and the power of pessimistic expectations to cause economic
downturns of which labor income bears the brunt. Importantly, the model can give rise to
a Smithian growth effect whereby a larger market leads to higher productivity through
increased division of labor.
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The most interesting, because the most distinctive, findings have to do with “the use of
knowledge in society.” First, the economy can be briefly devastated by an information
breakdown, but it is quite resilient. As long as the skill reputation system is operating
normally, it can regenerate lost information fairly quickly. But the system depends on
agents’ performance being reliably observable. If agents are sometimes able to conceal
their failures, the efficiency of the economy falls sharply. This suggests a reason for the
observed fact that people prefer to rely on networking (“weak ties”) to do business and to
hire. Networks create local ‘panopticons,’ where all events, including negative events,
are observable, where strategic concealment is impossible, and where, therefore, agents
are able to derive from induction accurate information about the quality of other agents’
skills. In short, networks may be the environments in which Bayesian skill reputation
systems can operate.
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CURRICULUM VITAE
Nathanael Smith graduated from Fairview High School, Boulder, Colorado, in 1996. He
received his Bachelor of Arts from Notre Dame in 2001. He received a Masters of Public
Administration / International Development from the Kennedy School of Government at
Harvard University in 2003. He was employed at the World Bank (2003-2004), the Cato
Institute (2004-2005), and again at the World Bank (2005-2007).
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